W&M ScholarWorks
Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects
1965
On Hereditary Properties of Topological Spaces
John Turner Conway College of William & Mary - Arts & Sciences
Follow this and additional works at: https://scholarworks.wm.edu/etd
Part of the Mathematics Commons
Recommended Citation Conway, John Turner, "On Hereditary Properties of Topological Spaces" (1965). Dissertations, Theses, and Masters Projects. Paper 1539624584. https://dx.doi.org/doi:10.21220/s2-h9tx-bw66
This Thesis is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. ON HEREDITARY PROPERTIES
OF TOPOLOGICAL SPACES
A Thesis
Presented to The Faculty of the Department of Mathematics
The College of William and Mary In Virginia
In Partial Fulfillment
Of the Requirements for the Degree of
Master of Arts
By John Turner Conway
August 1965 APPROVAL SHEET
This thesis Is submitted In partial fulfillment of
the requirements for the degree of
Master of Arts
1L ~ Author
Approved, July 1 9 6 5:
UJ. F. W. Weller, Ph.D. ““
S. H. Lawrence, Ph.D.
X . tl/kaJU s ______H* B* E asie r, M. S. ACKNOWLEDGMENTS i The author wishes to express his deepest gratitude to Professor
F. W. Weller under whose guidance this investigation was conducted.
The author is also indebted to Professor S. H. Lawrence and
Professor H. B. Easier for their careful reading and criticism of the manuscript. He wishes also to acknowledge the use of the facilities at the Langley Research Center where part of the research was undertaken.
H i TABLE OF COreEWrS
ACKHOWLEDGSMEBTS • • ...... • ...... • . . * « . i l l
ABSTRACT > • .• • '•. • • • • .• ■ « • • • * • ■ * ( •• • •" • • v
XlRROmCTIOV '-v*' V: /Vy-*- V'' i"* 2
STOOLS AID BOTATXOH v .... J lW TTMkk*fiTTARY PBQPEbTJJSS, ,#y ’o* ' - ■*.' 6 The ' T^ ---Spacer,..5. ; ** 10
The JBB>USd01?fif' Sj)£lC6 •**.,■■ '* V • ' #V. , • ••’ '• . « 12
The Regular and T^ Spaces • . . . . • « • . . . * . .. . • .* . • * 13 ^The.'Completely Regular andTIchonov Spaces . •,. • ,.,; • 15
The F i r s t and Second Axiom. Spaces • ^ . !>,.•;:*>,■■«,-.> 17
Metrlzahle; Space7 ■ .• ^ -y;v y * *• *-■•/23'
\..T >• i;*y 2**-
gv^. -y;. ■♦’Nl', :' 'ff' 1
Formal and Conrpletely Honaal Spaces . , ...... /■,* *. ;;*•>.•. > *<3
• ’ • _ * I' "*v V • V" "• '• .' • ♦ \ •• '. • v**.' '• < ,'■'••■ f : • 36
•• •• , • '• i»r ■ v*; -'• • • • 59
VITA #' '«> • « •' • • • • • ’ • . • ■ ■» • • " ' ■ /X,-Y ", ..r-" ■ 's=, :: '. ^ 'i.\'
I*Jb* ABSTRACT
A property of a topological space Is termed hereditary if and only if every subspace of a space with the property also has the property. In this thesis we determine some topological properties which are hereditary and investigate necessary and sufficient conditions for subspaces to possess properties of spaces which are not in general hereditary.
We show that the properties of being T^, Hausdorff, regular, T^, completely regular, Tichonov, first and second axiom, metrlzable, and completely normal spaces are hereditary. In addition ve prove that a subset of a subspace of a topological space is connected, compact, or Lindelof in the subspace if and only if it is connected, compact, or Lindelof in the space. We give examples to show that connected, separable, compact, locally compact, Lindelbf, paracompact, normal, and fully normal spaces are not hereditary, but prove that second axiom and separable metric spaces are hereditarily separable and that sub spaces formed from closed subsets of compact, Lindelof, paracompact, normal and fully normal spaces also possess the property. Further more subspaces formed from T q subsets (subsets formed from a countable union of closed subsets) of Lindelof and paracompact spaces also possess the property.
v OH HEREDITARY PROPERTIES
OF TOPOLOGICAL SPACES INTRODUCTION
A property of a topological space is termed hereditary if and only if every subspace of a space with the property also has the property. The purpose of this thesis is to determine some topological properties which are hereditary and to investigate necessary and sufficient conditions for subspaces to possess properties of spaces which are not in general hereditary.
We prove that the properties of being T^, Hausdorff, regular, T^, completely regular, Tichonov, first and second axiom, metrizable, and completely normal spaces axe hereditary. In addition we prove that a subset of a subspace of a topological space is connected, compact, or Lindelof in the subspace if and only if it is connected, compact, or Lindelof in the space.
We use the real line as an example of a connected space which has a subspace that is not connected. We show by an embedding defined by Pervin (l1*, p. 8U) that every topological space is a subspace of a separable space whence separability is not hereditary.
We prove, however, that every second axiom space and every separable metric space is hereditarily separable.
We use Alexandroff1 s theorem on one point compactification
(9 i P* 150) to show that every topological space is a subspace of a compact space whence compactness is not a hereditary property.
Since every compact space is locally compact and Lindelof these properties are not hereditary. Paracompactness is not hereditary as we can find a nonparacompact space whose one point compact!fication is paracompact. We prove that subspaces that are formed from closed subsets of the space are compact and that in a compact Hausdorff space, subspaces are compact if and only if they are formed from closed subsets. Furthermore we prove that if every open subset of a compact space forms a compact subspace, then all subspaces of the space are compact.
We use Fort's space (5) as another example of a Lindelof space which has a subspace that is not Lindelof, but we prove that closed subsets and F0 subsets (subsets formed from a countable union of closed subsets) form subspaces that are1 Lindelof. We show that if every subspace of a Lindelof space formed from an open subset is
Lindelof, then all subspaces of the space are Lindelof.
Although the property of paracompactness as defined by
Dieudonne is not hereditary, we include Dieudonn^'s proof (3) that every subspace of a paracompact space which is formed from a closed subset is paracompact, and if every open subset of a paracompact space forms a paracompact subspace, then all subspaces of the space are paracompact. Michael (12) extended this to show that every Fa subset of a paracompact space forms a paracompact subspace. We use Fervin's modification (14, p. 93) of the Tichonov example (9, p. 132) to show that normality is not a hereditary property. We prove that closed subsets of a normal space form normal subspaces and that a space is completely normal if and only if every subspace is normal. k
Stone (16) shoved that the property of paracompactness Is equivalent to a space being fully normal and T^, whence full normality is not a hereditary property. We prove that closed subsets of fully normal spaces form fully normal subspaces.
Although the details of theorems 1-12 of this paper were worked out by the author, discussion of these concepts can be found in most elementary topology texts such as entries (l), (6), (7)# (8), (10), (1*0, and (15) of the bibliography. The author was unable to find details of the remaining theorems, with the exception of those referenced in the paper, in the literature. SYMBOLS AND NOTATION
X a set of elements
0 the empty set
S i a family of sets xeX x is an element of the set X xfiX x is not an element of the set X
A C B the set A is contained in the set B
\( i6 a subfamily of the family
A U B the union of the sets A and B
AO B the intersection of the sets A and B
^ (A, B) the juncture of the set A and the set B
X a the closure of the set A <4 A the interior of the set A
8 A the boundary of the set A t A the complement of the set A
German l e t t e r s such as denote a fam ily of s e ts . In g en eral, the symbol X means a topological space with topology <£} , unless otherwise stated, and is written (X,jf>).
5 ON HEREDITARY PROPERTIES
Definition: Let $3 be a family of subsets of a nonempty set X such that,
(1) the union of the members of any subfamily of $? is a member o f £ ) , and (2) the in te rs e c tio n o f a f i n i t e number o f members of 5P i s a member of
Then P is termed a topology for X and the ordered pair (X, Jp ) is called a topological space. A subset 0 of X is open if and only if 0 is a member of P . A subset C of X is closed if and only if Example: (a) Let 0 be the. vacuous subfamily of J- Then, by definition (l^, p. 37) , ? (1) U G s 0 Ge d (2 ) n o = x Ge ejr so that the space X itself and the empty set 0 are open in any topology. Define £> =|x, 0}. 6 7 Then is called the indiscrete topology for X, and (X, 9P) is an indiscrete topological space, (h) Let he the family of all subsets of X , Then < ? is called the discrete topology for X, and (X, £>) is a discrete topological, space. In this paper we will sometimes denote for brevity a topological space (X, £> ) simply as X. Definition: Let (X, 9 ) be a topological space and let Z be a subset of X. The induced or relative topology for Z is the collection X H £>of all sets which are intersections of Z w ith members o f p > and we write zfl j Q: there exists 0€ We must first show that the induced topology is indeed a topology for Z, where Z Q X and (X, P) is a topological space. Theorem 1: Let (X, £>) be a topological space and Z Q X. <■ Then (Z, Zf\ {?) is a topological space where Z O there exists Oe Jp with Q » Z o|. Proof: (l) First we must show that ^.C Zfl implies U h e Z f l 9 . I f * 0, then U h » 0 = zO0 e Zn p He*v p 8 since $e 4? . Assume Then He implies there is an 0H e £ such that H = zPlOg and u h = u ( z n o H) * z n u o H€ z n -P He i\ He ft He sin ce U 0H e $ . He K Therefore U H eZ O ^. He S\. (2) How we must show that if Ji.£ Z HjP and 5\ is finite, then n H e zD £ . Be& I f SL = 4, then n h «z w znx e z r\$. He.fi. How assume ii / Then He it. implies there is an O^e ,P such that H = Z H 0„ and dh = n(zrioH) = znnoH eznjs He J\ He He fv 9 since n ° h 6 © - H6 when h . i s f i n i t e . T herefore O H e z f l f > Hefi. where fL Is finite. Hence (Z, zfl $D) is a topological space. We called a subset C of X, where (X, $p) is a topological space, closed if and only if the complement of C with respect to X is a member of $P. Thus a subset E of a subspace (Z, Z O §>) of (X, 5P) is closed if and only if the complement of E with respect to Z is a member of Zfl P. We will now show that the family of all closed subsets of Z with respect to the topology Z Pi is the same as the family formed by the intersection of Z with the subsets of X that are closed relative to the topology £ ). Theorem 2: Let (Z, Z O £>) be a subspa.ee of the topological space (X, *£>) • Let fe denote the family of closed subsets of X in (X, $P) and fe denote the family of closed subsets of Z in (Z, Z $p) . Then ZH fe* 6f. P roof: ( l) L et Ee G . Then there exists Q € z fl $3 such that E s ^ Q* Since Q e zO £>, there exists an 0 € such that Q a z Pi 0. Here 0 € - (zn cx z)u (zOCx'o ). $(j (zn cx o) » z HCjj o € zH &. (2) Bow let B s !fl C, Then there exists C e 6 such that E ■ zflc and CT AC ( S3, so that cz e -cz (znc) - zndx (znc) - (zndx z)U (zncx c) - 0 U(zHdx c) = zHCx c 6 z 0 3D. If we let Q - £„ E, then Q e Z f) & and Q - CL C„ E = E. Z Z <£• £i whence E € T?. The T^ - Space Definition: A topological space (X, $3) is a - space if and only if it satisfies the following axiom of Frechet: (Axiom Tj) If x and y are two distinct points of X, then there exist two open sets, one containing x but not Y> ^ the other containing y but not x. In the theorem which follows we give a simple characterization o f the T*l - spaces. Theorem 3: A topological space (X, SJD) is a T^ — space if and only if every subset of X consisting of exactly one point is closed. COLLEGE OF WILLIAM & MARY eeoe X, (X herefore T Hence and whence £>) , (x and ftf(x) xO a (x) - X e y x Then and X. in ts in o p t c tin is d hw b te hoe wih olws. follow which theorem the by shown f . hn x) n (g ae ebr f e wih mplies Here im which , . fe of members jjp in are are C(xg) (xg) and and C(x^) (x^) t. a th Then X. of 0 e y t a th such ? ej 0 ts is x e re e th plies im space - 3* a xaty oe nt lsd Lt ^ n g e stnctpoi ts in o p t c tin is d be Xg and x^ Let closed* s i t in o p one actly ex y y y “ h pr ry ofbig - pc is her t y, s s i as , ry ita d re e h s i space - a f o being erty p ro p The Proof: ( l ) Let ( X ,^ ) be a T^ - space w ith x and y y and x ith w space - T^ a be ) ,^ X ( Let ) l ( Proof: i * Then * 0 v (2) How suppose th a t every subset of X co n sistin g of of g sistin n co X of subset every t a th How suppose (2) Q ) i s a T^ - - space. T^ a s i ) X - (x) = U (y) c U 0y g X - (x ), ), (x - X g 0y U c (y) U = (x) - X Xg € £(x^) and x^ x^ and £(x^) € Xg 1 ^x) n Xg hnd ^(xg) € x1 - x = (x) - X x - - (x) / y/x y/x C(X x) e e . fe e )) (x - u y/x Oy e£> e£> Oy i i C3(Xg) * . 11 12 Theorem k: Let (X, £> be a T1 - sP*ce and Z be a subset of X. Then (Z, Z C\Q) i s a T^ - space. Proof: Let z^ and zg be distinct points of Z 9X. Since (X, jp ) is a T^ - space there exist sets 0^ and Og in such th a t z^ € 0^, Zg € Og and zg 4 0^, z^ 4 Og. Let Q1H z n o 1 and Qg s Z O.Og. Then Q^ and Qg are in Z H with z^ e Z Pi 0^ = Q1 and z^ 4 Og and Zg e Z H Og s Qg and Zg 4 01 , whence zx V ZPl Og a Qg and Zg 4 Z/1 01 s Qg. Therefore (Z, z n o is a T^ - space. The Hausdorff Space Definition: A topological space (X, ) is a Tg - space or Hausdorff space if and only if it satisfies the following axiom of Hausdorff: 13 (Tg axiom) If x and y are two distinct points of X, then there exist two disjoint open sets, one containing x and the other containing y. The property of being a Tg • space is hereditary, as is shown by the theorem which follows. Theorem 5 s Let (X,jp) be a Tg - space and Z be a subset of X. Then (Z, Z is a Tg - space. Proof: Let z Z g € Z 9 'X with z^ / Zg. Then (X,j|p ) a. Tg - space implies that there exist sets 0^, Og €J? *ith Z1 6 °1* Z2 € °2 and °i^ Og a Let Q1 - Z O 01 and Qg = Z H Og. Then Q^ and Qg are in Z with z^ e Z D 0^ s Q^, Zg e Z D Og s Qg and q2 (z n o^ n (zn Og) = z n (oxn o2) - z o 4 - 4. Therefore (Z, Z Hjp) is a Tg - space. The Regular and T^ Spaces Definition: A topological space (X, s ? ) is regular if and only if it satisfies the following axiom of Vietoris known as the - axiom: (T5 axiom) If C is a closed subset of X and x Is a point of X not in C, then there exist two disjoint open sets, one containing C and the other containing x. The property of regularity is hereditary, as is shown by the theorem which follows. Theorem 6: Let (X, Proof: Let fe be the family of sets in Z closed relative to ZPi Then by theorem 2, fe « zO fe where fe is the family o f s e ts in X closed r e la tiv e to SP. L et E e & and z € Cz E. Then E e fe = Z n fc implies that there exists C e & such that E « Z Pi C and Cz E a zOCx E a zDCx (zH c) a Z Pi (dx ZUCx C) a ZPl6 X C C C. How since C 6 and z 6 C, there exist 0 and 0„ € X ' z such that C Q 0, z € <1z and OPl 0w z a 0 since (X, *£>) is regular, Let QaZPlO and QzaZp|Oz. Then Q, Qz € Z Pi ?>w ith E - z O c S z P l O - Q and z c Z Pi 08 - Q* 15 and «hq,z = (zno) n(znoj z = 0 since onO z » 0. Therefore (Z, ZPljp) Is regular. Definition; A topological space (X, Si ) is a - space if and only if it is a regular space which is also a T-^ - space. The property of being a T^ - space is hereditary since regularity and the property of being a - space are hereditary. The Completely Regular and Ti chancy Spaces Definition: Let (X, 2 ) and (Y> ) be topological spaces. ^ func^iQn f with domain X and range in Y is a mapping of the set X into the set Y, written f : X -4 Y, such that if x e X with f(x) * y1 and f(x) a y^ then y1 * y2* We say that a function f is continuous at x e X if for _ o every H € f\. such that f (x q) € H there exists 0 € 52 containing x q such that f(x) e H for every x e 0. The function f is continuous on X if and only if f is continuous at every x in X. Definition: A topological space (X, jp ) is completely regular if and only if for every nonempty closed subset C of X and any point x of X not in C, there exists a continuous mapping f: X -»(o, i] such that f(x) = 0 and f(y) = 1 for every y e C. 1 6 Theorem 7: A topological space (X> jS? ) is completely regular if and only if for every point x e X and open set 0X € Q such that x € 0X there exists a continuous mapping f: X -> [o> l) such that f(x) s 0 and f(y) » 1 for every y £ CO . X. Proof: (l) Let x e X and 0 £ $ such that x e 0 . Then (Jo € & and x £ do , so that (X, £ ) completely regular A . JL implies that there exists a continuous mapping f: X l^J such that f(x) = 0 and f(y) = 1 for every y e C0X £ fe * (2) How let <£ £ C £ 6 and x £ &C be given. If we p u t 0 - dc then x e 0 e 5? there exists a continuous mapping f: X -* [0, £]. such that f(x) - 0 and f(y) = 1 for every y e dOx = d-dc = C whence (X, 5?) Is completely regular. Complete regularity is a hereditary property as will be shown by the following theorem. Theorem 8: Let (X, $£ ) be a completely regular topological space and Z C x. Then (Z, Z 052 ) is completely regular. Proof : Let E be a nonempty subset of Z which is closed relative to Z 052 , and let x € E = Z O E. Then E £ Z O fe by theorem 2, so that there exists C e fe with E * ZO C and x e dz E = dz (z O C) » z Odx (Z O c) = z o (dx Z(J Cx C.) = (zndx z) u (zndx c) = z n d x c, 17 which implies that x e ^ C. How (X, J? ) completely regular implies that there exists a cbntinuous mapping f : X -* [o, l] such that f(x) = 0 and f(y) = 1 for every y € C. Furthermore Z c X and EC C which implies that the continuous mapping f: x “* [?> l] is also f | Z : Z l] * such that f(x) = 0 and f(y) = 1 for every y e E 9 C. Therefore (Z, Z OS) is completely regular. Definition: A topological space (X, 2 ) is a Tjchonov space if and only if it is a completely regular space which is also a T^ - space. The property of being a Tichonov space is hereditary, since complete regularity and the property of being a T^ - space are hereditary. The F irst and Second Axiom Spaces Definition: Let X be a nonempty set, and let V i and ^ be families of subsets of X. Then g generates X.L o r % Is a base for if and only if is the collection of all subsets of X which are unions of members of *§ . Note that, for any collection , the collection U generated by will always contain the empty set as one of its element^ since "ji contains the vacuous subfamily. Definition: Let (X, $£ ) be a topological space, and for each x e X let *= |0 : 0 € 2 and x e 0|. Let be a 1 8 family of subsets of X. Then is a base at x if and only if ^ c S and for every 0 in j2 there is a B in X X such that B is a subset of 0 . Theorem 9» Let XX and be families of subsets of a nonempty set X such that is a base for XX * Then (X,J2 ) is a topological space if and only if (i) for each x € X there is a B g such th a t x g B, and (ii) if B-j^ and Bg are in and x e B1H Bg> then there is a B 6 such that x e B and B c Bi O Bg. Proof: (l) If (X,XX ) is a topological space, then X e which implies that there exists ^ C with X = U B U * B eg» * sin ce % is a base for XX • Hence x e X implies that there exists B € * s * such that x € B. Now let B^> Bg g *^> and x g B^O Bg. Then “|b a base for XX implies C X^C and Blf Bg € XX implies Bg eXX since XX i s a topology. Now 1 ^ 0 B2 e V( implies that there exists such that Bxn Bg - U B whence there exists B e tj£* B such that x e'B c H Bg. 19 (2) First we will show that V( — V( implies (J A eW . A € U * then U A » 0 € X*(. A € VC Now ass vane \ r ^ 0f. Then A € vr implies that there exists such that A = U B and B UA - U (UB) = UB « )«( • A € \C A eU * B e ^ B € £>A A € V(* Now we will show that if v (* is a finite subfamily of %±( then Pi A € X(. A e >n(* If V(* = then n a = X, and if x € X, then by (i) there Ae X\C* exists B € *& with x 6 B . x ^ x 20 Let 12> s {BA : x e X I •. Then x = u W5 U ®x = u » ^ x x e X x € X B e£* implies that x = uB • b € * Now suppose « { A^, Ag j. We will prove Ag e , whence by induction the conclusion follows. If A^n A2_ * 0, then A^H Ag e . Now assume A^f) Ag + 0> and let x e A^HAg. Then A^, A^ € \\( and a base for implies that there exist fi ^ fig ,C £ such that a U B and Ag = y B , whence there exist B <• B e f>2 B^, Bg e P such th a t x e A^ and x e Bg C Ag. Here x e B ^B g, so that by (ii) there exists Bx e 11 such that x € B^c B10 B2 C Ai n A2. If we put /Jj* = | bx : x € A1H Ag j. , then 21 Aj_n A2 = U W C (JBX = U B S^OAg xeJ^OAg xe^flij B € £ * and Pi Ag » CJ B € X\ ( • B € £ * Therefore (X,")n( ) is a topological space. Definition: Let (X, S i ) be a topological space. Then (X, ) is a first axiom space if and only if it satisfies the follow ing axiom known as th e f i r s t axiom of c o u n ta b ility : (First axiom) For every x in X there is a countable base a t x. The property of being a first axiom space is hereditary, as i s shown by th e theorem which follow s. Theorem 1 0 . Let (X, S i ) be a first axiom topological space and let 2 c X. Then (Z, Z f\2) is a first axiom space. Proof: Let z e Z c x* Then (X, $L ) first axiom implies that there exists a countable base 0 at z with respect to SL . Now 0 z 9 S i z implies Z Pi ^ z ^ Z fl £ z } so if H z € Z Pi jSL z , then there exists 0 € ££ such that H = zPlO. But 0 € SL z z z implies there exists B e such that B 90, whence there z exists Z Pi B e Z Pi 0 such that Z Pi B £ z Pi 0 = H . Furthermore * z z z n $ „ is countable, since ji is countable, z z 22 Therefore Z Pi is a countable base at z with respect to Z Pi 2*. Definition: Let (X, & ) be a topological space. Then (X, ) is a second axiom space if and only if there exists a countable base for . This is known as the second axiom of countability. The property of being a second axiom space is hereditary, as i s shown by th e theorem which fo llo w s. Theorem 11: Let (X,j£ ) be a second axiom topological space and let Z 9 X. Then (Z, Z 0 2 ) is a second axiom space. Proof: Let H € Z O S i > and let be a countable base for Then there is an 0 e such that H = Z O 0, and (X, ) second axiom implies there is a ** £ such th a t 0 » U B . Be** Thus H = Z O UB * U (ZOB). Also Z fl^is B B eiS * countable. Therefore Z fl ^ is a countable base for Z O & , and (Z, Z H 2 ) is a second axiom space. 23 Metrizable Space Definition: Let X be an arbitrary nonempty set, and let d be a nonnegative real-valued function defined on X x X. If for all x, y, and z in X, (1) x b y if and only if d(x, y) » 0 (2 ) d(x, y) = d(y, x) (3 ) d(x, z) < d(x, y) + d(y, z), then d is termed a metric for the set X. Definition: Let d be a metric defined on the set X. Let x be a point in X and r be a nonnegative real number. The set S(x, r) a |y : d(x, y) < rf is called the open sphere with center x and radius r . We shall use the symbol («* to denote the family of open spheres S(x, r) for all xeX and all real numbers r > o. Definition: Let X be a nonempty set with metric d. A topology for X can be obtained by using as a base the family ( 5 of open spheres. (7, p» 6 0 ). This topology is said to be induced by the metric dw We shall denote this topology by The pair (X, $£$) is called a metric topological space. Thus a metric space is a topological space whose topology is induced by the metric d. Definition: Let (X, $2. ) be a topological space. Then (X, & ) i s metrizable if and only if there exists a metric d on X x X such that the topology SL^ induced by d is identical with SL . The next theorem shows that the-property of being metrizable is hereditary. 2k Theorem 12: Let (X, ) be a topological space which is metrizable, and let Z ^ x . Then (Z, zO£>) is metrizable. Proof; Wow (X,£>) metrizable implies that there exists a metric d for X such that • Let f\. * zOJ>, ajid let be the topology induced on Z by the metric d|z. We will show = fi, whence (z, z n £>) i s m etrizab le. Suppose He f. and zeH. Wow He ft implies that there exists Oe Jjb such that H = zflO . Then zeH £ 0. Since Q ^ , there is an r > o and zQ eX such that ze S(zQ, r) c 0. Put H^ s zflS(z0, r) |z* ; z’eZ and d(z!, zQ) < rj-. Here is a sphere relative to Z. Then ze % e fta and %C ZflO = H. It follows that. f\ £ fi^. Wow let % e fifl, and let ze H^. Then there exist r > o and zQ eX such that zeZriS(z0, r) £ h^., where S(zQ, r) e . If we put H » zns(zQ, r) then ze He ft and H C H^. Hence C fi. Therefore and (Z, Z f) 0 ) is metrizable. Connected Space Definition: Let (X, £)) be a topological space and E £ X. Then E is separable into sets A and B» written E = a | b, provided there exist sets A, B eX such that (1) A and B are nonempty (2) E = AU B (3) y(A, B) = (AnXB)U(KAnB) = 0- The set E is connected if there do not exist sets A and B such that E is separable into A and B. A topological space (X, is said to be connected if and only if the set X is connected. 25 The example which follows shows that the property of being connected is not hereditary. Example: The real line R with the topology generated by the collection of all open intervals is connected ( 8, p . l*f) or ( 2 , p . 6*0 , but has a subspace that is not connected. Let A B (0, 1), B = (2, 5) and S = A U B. Then SCR and (i) A and B are nonempty subsets of S ( i i ) AUB=S (iii) >(A, B) = (AnXB)U(KAriS) = 0. Therefore S is not connected, and we can conclude that connectedness is not a hereditary property. Theorem 13: (14, p. 51) Let E be a subset of Z, where (Z, z C l p ) is a subspace of the topological space (X, jp). Then E is connected in Z if any only if E is connected in X. Proof: In order for E to not be connected there must be a separation of E into nonempty sets A and B such that A U B » E and ■i X a , b) = 0 . Since E C Z c X, we have A C Z and B Q Z, If there exist nonempty sets A and B with A {J B = E, then = (Afl Z HtC x B)U(KX Afl ZHB) = (AflK z B)U( K Z AflB), whence %(A, B) = 0 in Z if and only if ^(A, B) = 0 in X. Hence E is connected in . Z if and only if E is connected in X. 26 Separable Space Definition: Let (X, £3) be a topological apace and EQ X. Then E is dense in X if and only if XE «= X. Definition: Let (X, S3) be a topological space. Then (X, £3) is separable if and only if there exists a countable dense subset o f X. We will show that the property of being separable is not hereditary by showing that every topological space is a subspace of a separable topological space. (l4, p. 84). Theorem l4: Let (X, £P) be a topological space (in particular a nonseparable space). Let 00 be a point such that » X. Then X* « X U («) with the topology S3* • jo* : 0 *= 0 U (“) for 0 e is a separable topological space and (X> S3) is a subspace. Proof: First we will show that (X*, £3) is a topological space. (i) Let /L* — JP • If • $ then u Hi i c . H € (c j. ^ ^ Assume f\^ £ 0 and A. » \ 0 : 0 € jp such that 0 (J (°°) * 0 for 0* e Then U o*« U(0 U(“)) « UO U (“) e since U 0 ejP . ' 0 s ft* 0 eh . 0 s JL 0 e k 27 (ii) Now suppose fu* C <£)*with ft* finite and 0 i fvT • If ft* = 0 then H O = X* e £> *. H e Otherwise n o* - n (ouw) - ( n o)u(-) ^ $>*, 0 € ft* O'e ft OefL since n 0 e p 0 € ft as finite implies ft is finite also. Therefore (X*, P*) is a topological space. Now we will show that (X*, is separable. Let 0 * e Then (<») € 0 *, since 0 * = 0 U (») where 0 e p . Let x e X*. Then for every 0 * such that x e 0* we have * n (°°) * (») and x e whence X* «^(«>). Thus Xt00) - X*, and )<»[ ^ ^ _rj_ is a countable dense subset of X . Therefore (X , p ) is separable. Clearly (X, f P ) is a subspace of (X*, £$ ), since X Cx* = X U W and X n & - ) x O 0 * : 0 * 6 £>*} = j x n ( 0 U '(» )) ; 0 £ p | » jxAo : 0 e £D}=)0:0e 9P}= p- 2 8 For another example of a separable space that is not hereditary see Sail and Spencer (7, p. 1 1 2 , prob. 53). Theorem 15: (15 > P* 60) In every second axiom topological space separability is hereditary. Proof; Since each sub space of a second axiom space is also ( a second axiom space, we need to show that every second axiom space is separable. Now (X, ) second axiom implies that there is a countable base for Let “^5 = J B^ ; n is a positive integer}. Choose e 3^. We may assume n > 1 implies that ^ 0. Put 00 E = U (%). n=l Clearly E is a countable subset of X. Now l e t x e X and •£> x = )B : B e "Jj and x € B}« Here £ X iB a base 8t x B 6 P X that there exists an n such that B * Bn, whence Xn e BnDE = BpIE so th a t B Pi E ^ 0. Thus x e VC E, end X Q VC E . Therefore VC E = X, and E is a countable dense subset of X, whence (X, p ) i s separable • 29 Theorem 1 6 : Every separable metric space is hereditarily separable. Proof; This follows immediately from theorem 15 and the fact that a metric space is separable if and only if it is second axiom. ( 6 , p. 1 2 1 ). Compact and Locally Compact Spaces Definition: A family of sets ^ is an open cover of the set E if and only if each G in eg is open and E £ U G. G e d If there is a finite subfamily C S* of 0? such that E £ U G G € Definition: A subset E of a topological space (X, jp ) is compact in X if and only if every open covering of E is reducible to a finite subcovering. A topological space (X, P ) is compact if and only if X is compact. Definition: A set A in a topological space (X, ) is a neighborhood of a point x € X if and only if A contains an 4 open set to which x belongs. A topological space is locally compact if and only if each point of the space has a compact neighborhood. Clearly every compact space (X, ) is locally compact, since X is a compact neighborhood of each of its p o in ts. 30 Definition: The one -point compactification of a topological space (X, jp ) is the set X* ■ X U (°°) where « is any point * not a member o f X w ith th e topology £3 whose members: are a l l the sets in p together with all subsets G of X such that X - G = C *G is a closed and compact subset of X. X We w ill state here without proof Alexandroff * s theorem which shows that compactness is not hereditary, since every noncompact space can be embedded in a compact space. For a proof see Kelley (9, p. 150). Theorem 17 (Alexandroff): The one point compactification (x*, e f ) of a topological space (X, §D ) is compact and (X, < p ) is a subspace. The space (X*, P*>. is Hausdorff if and only if X is locally compact and Hausdorff. Local compactness is not hereditary* since the one point compactification of a space which is not locally compact is compact and hence locally compact* Theorem 18: (l^, p. 57) Let £ be a subset of Z where (Z, Z n p>) is a subspace of the topological space (X, £3). Then E is compact in Z if and only if E is compact in X. Proof: (l) Suppose E is compact in X* Let ft £ Zfl £3 such th a t E £ U H. H € a Here H € fc. implies that there is an 0g e such th a t 31 H = z f l Ojj. Then e C u h = U (znow)= z n u o TC(j cu H € (v H e *. H H c H e and E compact in X implies that there is a finite fc* Q ft. such t h a t E C y o„. H € H But E C Z also, so that Ec z n U oh = U (^0%) = y h, H € k* H ek* H ef* end E is compact in Z. (2) Now suppose E is compact in Z* Let (S — such t h a t EC(J G G e (5 and define F- s {h : there exists G e (3 such t h a t H = zriG(. Since and e £ Z, we have E C Z n u G = U (zH g) = U H. G e <5 G e Cs Heft- Now R $ Z O D and E is compact in Z, so that there exists a finite subfamily f\* Q fu such that E £ U H. H e 32 Here H £ implies that there is a Gg € & such th a t H = Z n gh. Define <** s : ® € & *} • Then <5 - * is a finite subfamily of Cs since A* is finite, and e £ uh - u(z n %) - z n ugh * z n ug c u g , He A* Heft* He A* G e(5* G ec£* and E is compact in X. Hence compactness is independent of the subspace. Although compactness is not in general hereditary, the next theorem shows that every closed subset of a compact space is compact. Theorem 19: (U, p. 51) Let F be a closed subset of the compact topological space (X, $?). Then F is compact in X and ( f , f nj?) is compact. Proof: Let Cf be an open cover of F. Here F is closed, so d F is open. Define A to be all of the sets in (jr together with £f. Then A S j? and X = FU CF £ UG U = UH , G eCs H e f t 33 so there exists a finite subfamily Ji* c A such that XC (JH E € ft* since X is compact. Define d * n ft* - (dp). Then P = PA* £ FA (JH He#* = (FA UH) U(F PI CJF) He#* a # dp = FO uo £ Uo , G « d* oe d * whence F is compact in X. From theorem 1 8 we also know that F is compact in (F,P fi )., whence (PjPflJ) is a compact subspace. Theorem 20: (ll, p. 5 1 ) Let E be a subset of a compact, Hausdorff space (X, ). Then E is compact in X if and only if E is closed. Proof: From theorem 19 we know that closed subsets of a com p act space are compact, so i t rem ains to show th a t E Compact in X implies that E is closed. Suppose y is fixed and y e C?E. We will exhibit an 0 e $ such th a t y € 0 C £E, whence x € 0 **, y € Qx” and 0 X* H Ox” = since (X, ) is Hausdorff. Nov E * U(x) c u o x', X € E X € E so th e re e x is ts fx«lWL such th a t E C U 0 , ' . 3* L J Define Then 0 e x and y e 0. Also 1 J J J n Implies O x j ' n v = i , so J o- " c do*. • XJ xj and 0 S ,0, °x " ^ P 'CO ' = <*U 0 • C dE. J=1 J J=1 J j«l J Therefore E is closed, since dE is open. We are also able to shov in the next theorem that if every open subset of a compact space is compact, then every, subset is compact and compactness is hereditary. 55 Theorem 21: Let Z be a subset of a compact space (X, ). If every open subset of X is compact in X, then Z is compact in X and (Z,Z H j?) is compact. P roof: Let Z ^ X and (5 be an open cover of Z. Put U = UO € $. 0 e d Then there exists a finite subfamily of d> such th a t V C U O , 0 e since U is compact. Then ZC uC (JO , and (£* is a finite 0 e (s* subcover of Z. Thus Z is compact in X, and by theorem 18, (Z,Zfl S?) is compact. D e fin itio n : Let (X, %) be a topological space. An subset of X is a set formed from a countable union of closed subsets of (X, ^). The example which follows shows that P^ subsets of a com pact space are not necessarily compact. Example: Consider the real line R with the topology gen erated by the collection of all open intervals of real numbers. R is an Fff set, since we can write 09 R * U [-n ,n j. n=i The one point compactification of R is compact, and R is an 36 Fa subset ( 9, p. 1^9). However R is not compact, since 00 R 9 U (-n,n) n=l and there is no finite subcover of this cover. The Lindelof Space Definition: A subset E of a topological space (X, jj) is Lindelof in X if and only if every open cover of E is reducible to a countable subcover. A topological space (X, is Lindelof if and only if X is Lindelof. Theorem 22: Let i? be a subset of Z and (Z,ZH 5?) be a subspace of the topological space (X, $?). Then E is Lindelof in Z if and only if Z is Lindelof in X. Proof: The proof of this theorem is identical to the proof of theorem 1 8 with the words "compact" and "finite” replaced by the words "Lindelof” and "countable" respectively. Hence the Lindelof property is independent of the subspace. Definition: Let X be an uncountable set and let p be a fixed point of X. Let be the family of subsets A such that either (i) p / A or (ii) p c A and <*A is finite. Then (X,\n() i s known as F o r tfs space. (5)* Lemma: Fort's space is a topological space. 37 Proof : Since p i 0, 0 c V( end X c\J^ since £x = 0 Is f i n i t e . (1) First we will show that (JO c for £.V(* G c Cs (i) Suppose for every G c Cs th a t p i G. Then p t J G , G c (5 and (JG c VL- Gc <& (ii) Suppose there exists Gp c Cs such that p € Gp. Then GpC (JG so (*U0 — which is finite. Thus G c Cs G c<£ * p c (JG and dUG is finite, so that (JG €\^. GcGr G c CS G c (j> (2) Now suppose <& S V ( and Ck is finite. ( i ) I f p 4 G for some G c (£ then p i flG and G c CS flG € G € Cs (ii) Now suppose p c G for every GcC§ so pc fiG . G C dr Since Ck is finite we can write & s (G..)n . Then n n d C \0 - S O 6 < * U C Gv But (5.G* is finite since O e( 5 j= l 3 J«a 3 00 p c G- fo r j * 1, 2, , . . , n. Thus (J (? Gj i s n=l finite and O G c G c tf Therefore ( X, X»() is a topological space. The theorem which follows shows that Fort's space is an example of a Lindelof space which is not hereditarily Lindelof. As another 38 example we could take the one point compactification of any non- Lindelof space* The resulting space Is compact and thus Lindelof. Theorem 23: Fort's space (X,\JJ is compact and hence Lindelof, but X - (p) Is a subspace which is not Lindelof. Proof: Let CS £V(. such that X = UG. G€ C9 Since p € X there exists Gp e & such th a t p € v whence £Gp is finite. Now x € dCL C X - U G ^ G€ & Implies there exists Gx € Cs such that x e Gx. Then X - Gp U ClCL-SGpU UG* , x eC?Gp which Is a finite subcover of X since Is finite. Hence (X,\r{) Is compact and thus Lindelof. Now consider the subset X - (p) of X. If x e X - (p), then (x) e since p i (x). Hence the sets (x) for x € X - (p) form an open cover of X - (p), but there exists no countable subcover since the points In X are uncountable. Thus X - (p) is not Lindelof in X and from theorem 22, the sub space (X - (p), (X - (p))nU ) is not Lindelof. Theorem 2k: Let P be a closed subset of a Lindelof space (X, $?). Then P is Lindelof in X, and (P,P O $) i «9 a Lindelof space. Proof: This proof that P is Lindelof in X is identical to the proof of theorem 19 with the words "finite” and "compact" replaced by the words "countable" and. "Lindelof," respectively. By theorem 22, the Lindelof property is independent of the sub space so (F,F H 5?) is Lindelof. Theorem 25: Let (X, j?) be a Lindelof space and Z be a Fa subset of X. Then Z is Lindelof in X, and (Z#Zfl 5?) is a Lindelof space. Proof: Let <& be an open cover o f Z. Then Z £ U G . G ec£ Since Z is an Pff subset of X, t Z » u o±, i* l where each is closed in X and t % «. By theorem 2k, each C±, f o r i « 1 , 2, • • ., is Lindelof in X, since each is ko closed. Also for each i, t Ci c u C. * Z £ u o , j= i J o e 6 , '* V so that there is a countable subfamily (£^* o f c£ such th a t c± e'. ug • Gc d ±* Let (3* = | g : G e for i * 1, 2, . . .} . Then t t Z = U q c u (UG) = UG, J/t-; 1 1 1 1 G€ C#i* G €(s* and CS* is a countable subcollection of which covers Z. Thus Z is Lindelof in X and by theorem 2 2 , ( Z, Z fY $?) i s a Lindelof space. Theorem 26: Let (X, 5?) ^ a Lindelof space. If every open subset of X is Lindelof in X, then every subset Z of X is Lindelof in X and (X, 5?) is hereditarily Lindelof. Proof: The proof that Z is LindelBf in X is identical to the proof of theorem 21 with the words "finite” and "compact" replaced by the words "countable" and "Lindelof," respectively. Thus Z is Lindelof in X and by theorem 22, (Z,Zfl P ) i s L indelof. k l Theorem 27: (l^, p. 8l) Every second axiom space is hereditarily Lindelof. Proof: Let E he a subset of a second axiom space (X, jj), and let Cs be an open cover of E. Then E £ U G . G 6 C5 Since (X, J ? ) is second axiom, there exists » s a countable base for $?. Define J s |n : there exists G e Cs such that C g}. Now n € Jimplies that there is a G = Gn € CST such that L et P* 3 [Bq : n e j} and CS* s {^h : n € J }“ Here is countable which implies that c£* is a countable subfamily of Cs since ^3* £ which is countable. Now x € E £ UG c G € cfc implies that there is a G e (} C J such that x € G, and lj-2 is a "base for £ so there exists Bq € such that x 6 BjjC G -whence n « J im plies x € (JBn* n € J Thus EE UBn CUGn * UG C UG n € J n € J GeCy* G c ^ so C$* is a countable subcover of Cs. Therefore E is Lindelof in X -whence (E,E H $ ) is Lindelof. If we put E * X in the above proof we have (X, 5?) second axiom im plies (X, J?) is L indelof. I t can be shown th a t a m etric space (X, J?) i s a L indelof space if and only if it is a separable space. ( 6 , p. 123). Furthermore in a metric space (X, J?) that is also Lindelof, the Lindelof property is hereditary, since separability is hereditary in a metric space by theorem 1 5* Definition: A topological space (X, J?) is said to have the Lindelof* property if and only if for every subfamily of 5? there exists a countable subfamily (£* of (J such that U G = U G . G € G G €($* Theorem 28: A topological space (X, J?) is hereditarily Lindelof if and only if (X, 5?) has the Lindelof* property. *3 Proof: (l) Suppose (X, j?) has the Lindelof* property and Z c x. Let Cs £ & such that CS* is a cover of Z, that is Z E U G . G € (3* Then there exists a countable subfamily C?* of (5 such th a t u g = U g G e CS* G s G* since (X, $?) has the Lindelof* property. Here Z £ U G - U G , G€C^ G€(S* whence Z is Lindelof in X. Furthermore Z is Lindelof in Z, th a t i s (Z ,zn S?) is Lindelof, by theorem 2 2 . Therefore, (X, $?) is hereditarily Lindelof. (2) Now suppose (X, $?) is hereditarily Lindelof. Let and put Z s U G . G € Cs Then (Z,Zp) ) is Lindelof, whence by theorem 22, Z is Lindelof in X. This implies that there is a countable subfamily {£* o f ( s such th a t UG « Z C UG . GeCr g sCr* But U G C U G , G € C5* G € Cs* 1+4 whence U % - U o a e <3 Oeg and (X, £3 ) has the Lindelof* property. Paracompact Space Definition: Let (X, J[b ) he a topological space and let and he open covers of X* Then VC is termed a refinement of if and only if for each A € there is a B € *^ such that A is a subset of B. Definition: A cover V\(^ of a topological space X is termed locally finite if and only if for each point x of X there is a neighborhood containing x which intersects only a finite number of sets of Definition: A subset E of a Hausdorff space (X, $3) is termed paracompact in X if and only if every open covering of E has an open locally finite refinement which covers E. A topological space (X, ) is paracompact if and only if X is paracompact. The property of being paracompact is not hereditary, as can be seen by taking the one point compactification of a space triiich is not paracompact but Is locally compact and Hausdorff (9> P* 165 and 172) • Dieudonne ( 3 ) showed that every closed subset of a paracompact space is paracompact. Michael (12) extended this result by showing that every F subset of a paracompact a space i s paracom pact. Dowker (U) gives fu rth e r d iscu ssio n of paracompactness not being hereditary. Theorem 29: (3) Let F be a closed subset of a paracompact space (X, £>). Then (F, F O jp) is paracompact. Proof: Let f \ be an open cover of F, where H € im plies H eF fl p . Then F C (j h . H € K Let <3 s j 6 : G e p and there exists H € fL such that H = Ffl Gf. Since F e 6 , GtF e £) and Cs together with j tt F | is an open cover of X. Let V C = iG : G cCs j U \ 0 F \ . Then X paracompact implies that there is a locally finite refinement V(* of (• Let Then A e V C * and F f l A J 0 implies that there Isa G e Cs such that A Q G and f H a C FflG - H e fv .; Since VC* is locally finite, for x € X there exists 0 € such th a t 0 intersects only a finite number 0 f sets in \\(* . But this implies that FflO intersects only a finite number k6 of sets in Furthermore p = fOx c u a, = FD u a = U (fHa)c (j H AeU* AeU* AeU* H € a H f 49 a P I f 4 9 whence fv. is a locally finite refinement of which covers F. Thus (F, F fi J)) is paracompact since by theorem 5, (X, ) Hausdorff implies (Z, Z(1 ) is Hausdorff. Corollary: Every closed subset of a paracompact space (X, £2) ) is paracompact in X* Proof: Let F be a closed subset of X and ci — such that C5 is an open cover of F. Here O F e and ft = {G : G e (s } U | OF } is an open cover of X which has a locally finite refinement ft* since X is paracompact. Then locally finite refinement of © which covers F, since F = PH x £ f H U H = F O U H =FflU « C U 6 • He/v* He fu* 0 e locally finite. V7 Theorem 50: (3) If every open subset of a paracompact space (X, Sb ) is paracompact and Z is any subset of X, then (Z, Z Pb £>) is paracompact. Proof: Let <5 be an open cover of Z such that 6 £ implies that there exists 0 € £> such that G = zO o. Let >*( = {A : A e £5 and there exists G e Cs such that G = Z H A } . Put U A. A € Then fo r each A £ V*( > UfjA s A,' so is an open cover of U with respect to U O . Hence there exists an open locally finite refinement ^(* of . Each A in Xx(* is open in 5b as well as U O ft-3 and covers Ze Then Z D a Q z H A ’ = 0 € (4 . A € A1 £ \>\t. Also z = zDx^z n uru = u (znA) = u g A e U * A e X ( * o e <3 * and * locally finite Implies Cs'* is locally finite. Thus (Z, Z n 5p) is paracompact. Normal and Completely Normal Spaces Definition: A topological space (X, ) is normal if and only if it satisfies the following axiom of Urysohn known as the T^ axiom: (T^ axiom) If C 1 and Gg are two disjoint closed subsets of X, then there exist two disjoint open sets, one containing and the other containing C^. Definition: A topological space (X, ) is completely nonssi if and only if for every two subsets ^ and A 2 o f X, such th a t Ag) = (A^ o XAg) u (K*10 a2) = 4, then there exist two disjoint open sets, one containing A^ and the other containing Ag. Every completely normal space (X, Jp ) is nomal, for if and Cg are two closed subsets which are disjoint, then % cx, c2) = 0, and there exist tvo disjoint open sets 0^ and 0 2 such th a t c 0^, C2 C02 and ( ^ 0 0 2 = 0. Theorem 31: Let Z be a subset of a completely normal space (X, £> ). Then (Z, Z fl $p ) is completely normal. Proof: Let A^ and Ag be subsets of Z such that 9Z ( A ^ Ag) = (A^ n X z Ag) U ( -Kz Aj^n Ag) = 0 . k9 Then (ax n kz a2) u(kz a1ha2) = (Aj^n z n ^ a2)u (zn kx a1Ha2) - n h x a 2 ) u ( k x A1 n a 2 ) = ^ L ' A2 )- And (X, SfD ) completely normal implies that there exist 0^ and 0 2 in & such th a t *1 - °1» *2 £ °2 Oxno2 = 0. But now AgC z f l 0 2 and ( z n c ^ ) n (Z n02) = 0 whence (Z} Z fl' P ) is completely normal. Theorem 32: (l^> P» 92) A topological space (X, 5p ) is completely normal if and only if every subspace of (X, JD ) is norm al. Proof; Let Z 5 X. We have already shown that complete normality is hereditary and that complete normality implies normality, whence (Z, Z O CD) is normal if (X, ) is completely normal. Now suppose that every subspace (Z, z fl £>) for Z Q X is normal. Let A^ and A^ be subsets of ~ X such that ^ (A j/ Ag) « 0 . 50 Consider the open set c< x a1 n ka2) = z * * * as a subspace of X. Then Z is normal with Z H VC/L^ and Z O XAg disjoint closed subsets of Z, sin ce (z* n XA^n (z* n ^ka^ = (d ( x a 1 n x a 2) 0 ^10 (d ( x^ n *a2) hxa2) = « tUA1 u c ka2) n r\(( c.xa1 u c k a 2) dxa2) = (x a 1 n c k az) 0( ha2 n d x a1) = 0. Thus (Z, zfl J)) normal implies that there exist disjoint sets % % *» Z* n XAj^C and Z* H KA 2 C G ^ . Now G^, s Z Pi implies that there are sets 0 ^ , Og € $5 such that G^ = Z H 0^ and G^ » Z H Og. But G^ and G^ are * _ in since Z £ , whence a1c z*n Xi^Ca^ c £> and Ag C Z* n X Ag c q € p . Hence (X, P ) is completely normal. 51 We w ill now give Pervln's example of a normal space which Is not completely normal and hence not hereditarily normal • Pervln (lU , p. 9 3 ) states that this Is a modification of the Tichonov example. ( 9, p. 132). Hall and Spencer (7, P* 2 9 1) also give an example. Example: Let (X^, ) be an uncountable discrete space (that is, every subset of X^ belongs to and (Xg, be an infinite discrete space. Clearly (X^, and (Xg, ^g) are Hausdorff since every subset of X^ belongs to and every subset of belongs to 2 * Furthermore these spaces are locally compact since each point of either space is contained in a compact neighborhood, namely the neighborhood consisting of the point i t s e l f . Let V = Xx (J (a) and Xg* = X 2 (J (0) be the one point compactifications of (X^, and (Xg, ^ 2)9 respectively. Then by Alexandroff' s theorem (X^, Sp^) and and (Xg, Jpg' 8X6 cofflPac"b and Hausdorff. * The cartesian product of X^ and Xg* is the set * * It < * * ) X « X^ x Xg * Xg) : x^ e X 1 and Xg e Xg >. 52 The set X together with the family, , formed by using as a base the collection of all sets 0 ^ x Og where 0 ^ € 02 e 5^2* is a ‘topological space, (l^, p. 129). Furthermore (X, %D ) is compact and Hausdorff since the cartesian product of Hausdorff spaces in Hausdorff ( 9, p. 9 2) and the cartesian product of compact spaces is compact. ( 9> p* 1^5 )• Therefore (X, §2 ) is normal, (l^, P«9l)« However the subspace X - {(a, p)} is not normal since the sets A * j(a, y) : y e Xg j. and B * j(x, 3) : x € X^ are disjoint closed sets in X - | (a, 3) } which are not contained in any disjoint open sets in X - j (a, $)}• Clearly A H B « 0 since 3 t Xg and a i X ^ A » {(x, y) : x ^ a X - {(a,*)}. and y € X2 } U {(x , 3 ) : x ^ a| = { (x, y) : x € X1 and y e Xg} = X ^ X * 53 which is open in X - { (a, 0) | . Therefore A is closed in X - {(a, 0) J . Similarily B is closed in X - {(a, 0)}. Now suppose there exist two open sets 0. and 0_ in A £ X - {(a, 0)} such that A? 0^ and B C 0g. We will show that 0A and 0^ cannot be disjoint. From our definition of each point set {(x, y)} where x ^ a and y ^ 0 is in <33 since it is the product of (x)e and (y) e ^ 2 * s *nce we took the one point compactification of discrete spaces X^ * and Xg, an open set in containing a must contain all but at most a finite number of points of X^, and an open set in * Xg containing 0 must contain all but at most a finite number of points of X2* It follows that, for each fixed y € Xg, a neighborhood of the point (a, y) must contain all but at most a finite number of the points (x, y) with x e X^. Similarily, for each fixed x e X1, a neighborhood of the point (x, 0 ) must contain all but at most a finite number of the points (x, y) with y € x2. Since Xg is infinite, choose a sequence y^, y2, ... of distinct points from it. The points (a, y^), (a, yg), ... are then distinct points of A and 0A must contain all but a finite number of points in each set { (x, y^) : x e | where y^ is one of the terms in the sequence. Now consider Og which contains B. Only a finite number of sets of the form * |fx, y) : y € x2 } 5U for x a fixed point of X^ can be contained in 0^ for otherwise there is an infinite sequence { j(x , 7j) : x € X1| . Furthermore at most a finite number of sets of the form j(x , y) : y € ‘* 2 } can be contained, except for exactly one point each, in 0g> fo r otherw ise 0A would not contain all but a finite number of points in both {(x, y1) : x € Xx \ and {(x, y2) : x e X2 y By induction, for n = 1, 2 > ..., only a finite number of sets of- the form {(x, y) : y e X*} can be contained, except for exactly n points each, in 0 ^ , fo r otherw ise 0^ would not contain all but a finite number of points, in the sets {(x, y±) : x € X1j fo r i m 1, 2 , ..., n + 1* Thus for only a countable number of 55 points x c X 1 can 0^ contain all but a finite number of points in each set {(x, y) : y e X* } . But this is a contradiction since for B C 0_ , 0^ would have to *“" B Jo satisfy this condition for each point (x, 0) € B and there are uncountably many such p o in ts . T herefore X - j (a , p)} i s not a normal subspace. Theorem 35 z Let Z be a closed subset of a normal space (X, gD ). Then (Z, Z Pi $p) is normal. Proof: Let E^ and E 2 be two disjoint subsets of Z that are closed relative to Z O £>. Then there exist C^, C 2 e & such th a t Ex = Z O C± € f* and E2 = Z H C2 e since Z is closed. Since (X, *£? ) is normal there exist Q£, 0 ’ € 5P such that ^ = Z fl Cl — °l Eg = Z ,0 Cg C 02 and ( I p l 0£ = 0. Here ^ = Z n C ^ z f l o ' s z O jl and = Z H ^ C Z Pi 0 £ e Z fl S3 and (zOo£) n (zn op = zrKo^nop = 0. Therefore (Z, Z fl S5) ia normal. 56 Fully Normal Space Definition: Let X be a nonempty set and be a collection of subsets of X. If x € X then the star at x of , denoted S(x, *)vxC ), is the union of the members of to which x * belongs. A collection of sets is a star refinement of a collection if and only if the family of stars of at points of X, called the star of , is a refinement of ^ . These concepts were introduced by J. W. Tukey (17, p. 53). Definition: A topological space (X, ^ ) is fully normal if and only if each open cover of X has an open star refinement. A. H. Stone ( l 6 ) showed that every fully normal - space is paracompact, and conversely that every paracompact space is fully normal (and T^). Thus the property of a space being fully normal is not hereditary, since the one point compactification of an arbitrary space (in particular one that is not fully normal) is paracompact and therefore fully normal. The theorem which follows shows that subspaces of fully normal spaces formed from closed subsets are fully normal. Theorem 3b: Let Z be a closed subset of a fully normal space (X, 5P). Then (Z, Z H %>) is fuHy normal. Proof: Let ijj> be a cover of Z which is open in Z fl P * Then for each B in ^ there exists 0 in such that B sZ H O ezH P . 57 Let jo : G e 'p and there is a B & such that B = Z PIG | (J |<±x Z} . Here Z € fe implies that (J,„ Z s . Let x e Z. Since is a covering of Z there exists B e with x e B e Z n sp. whence there is a G e such that x e G. Then i x = z(JdYzC u g u z G € ^ X and Cs is an open cover of X. Since (X, *5P) is fully normal there is an open star refinement ft of (i . Let \ t s |A : A = ZflH for H e {1}. Then A e X»( implies that there is a He Q ^3 such that A = Z f ) H whence A s Z Pi P and z e Z £ X 9 U H H e f t ao there is a H e ft with z e B and z t Z p) H. Thus \y(. is a cover of Z which is open relative to Z Pi SP . 58 We will show that X,( is a star refinement of ... Consider S(z, ) where z e Z. We assert that there exists B e such that S(z, Q. B. Here there exists G e & such that S(z, Then there exists e X^( such th a t w e A^ and z e A . Furthermore there exists H e fi_ such th a t w w Aw = Z H H w . Then z e Aw £ H w im plies H W £ s(z, ft )C G. T herefore and w € B. BIBLIOGRAPHY Calms, Stewart Scott. Introductory Topology* Hew York: The Ronald Press Co., 1 9 6 1. Bieudonne, J. Foundations of Modem Analysis, New. York: £ Academic Press Inc., i 9 6 0. 0 - * * Dieudonne , J. "Une Generalization des Eepaces Compacts." J. Math. Pures Appl. Vol. 23 (19^), 6 5-7 6. Dowker, C. H. "An Imbedding Theorem for Paracompact Metric S paces." Duke Math. J . Vol. Ik (19^7 ) y 6 3 9-6U5 . Fort, M. K., Jr. "Nested Neighborhoods in a Hausdorff Space" (Solution to problem 1*5 7 7 ) • Amer. Math. Monthly Vol. 62 (1955), 372. Gaal, Steven A. Point Set Topology. New York: Academic Press Inc., 1961*. Hall, Dick Wick, and Guilford L. Spencer II. Elementary Topology. New York: John Wiley and Sons, Inc., 1955 • Hocking, John G., and Gail S. Young. Topology. Reading, Mass. Addison-Wesley Publishing Co., Inc., 1 9 6 1. Kelley, John L. General Topology. Princeton, N. J.: D. Van N ostrand C o., I n c ., 1955* Lefschetz, Solomon. Introduction to Topology. Princeton, N. J Princeton University Press, 19^9• McShane, Edward James, and Truman Arthur Botts» Real Analysis. P rin c eto n , N. J . : D. Van N ostrand C o., I n c ., 1959* 60 12. Michael, Ernest. "A Note on Paracompact Spaces.” Proc. Amer. Math. Soc. Vol. k (1955)> 8 3 1-8 3 8. 13* Michael, Ernest. "Topologies on Spaces of Subsets.” Trans. Amer. Math. Soc. Vol. 71 (1951), 151-182. lU. Pervln, William J. Foundations of General Topology. New York: Academic Press Inc., 196 k. 15- Sierpinski, Waclav. General Topology, trans. C. Cecilia Krieger. Toronto: University of Toronto Press, 1952. 16. Stone, A. H. "Paracompactness and Product Spaces." B u ll. Amer. Math. Soc. Vol. 5 k (19^8), 977-982. 17. Tukey, J. W. "Convergence and Uniformity in Topology." Ann, of Math. Studies. Vol. 2, 19*10. 6 l VITA John Turner Conway Bom in Bradenton, Florida, February 27, 19k). Graduated from Citrus High School in Inverness, Florida, June 1958; B.S., Florida State University, June 2, 1 9 6 2. Entered on duty as a mathematician at the Langley Research Center of the Rational Aeronautics and Space Administration, Hampton, Virginia, J u ly 1 9 6 2. In September 1 9 6 3, the author entered the College of William and Mary as a graduate student in the Department of Mathematics.