SEPARATION AXIOMS and MINIMAL TOPOLOGIES by Saw-Ker Liaw B
SEPARATION AXIOMS AND MINIMAL TOPOLOGIES
by
Saw-Ker Liaw
B.Sc, Nanyang University, Singapore, 1967
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS
in the Department
of
MATHEMATICS
We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA
September, 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.
I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of HATH E MhHC £
The University of British Columbia Vancouver 8, Canada Abstract
A hierarchy of separation axioms can be obtained by considering which axiom implies another. This thesis studies the properties of some
separation axioms between TQ and and investigates where each of the axioms belongs in this hierarchy. The behaviours of the axioms under strengthenings of topologies and cartesian products are considered.
Given a set X, the family of all topologies defined on X is a complete lattice. A study of topologies which are minimal in this lattice with respect to a certain separation axiom is made. We consider certain such minimal spaces, obtain some characterizations and study some of their properties. Acknowledgement
I am greatly indebted to Professor T. Cramer, without whose guidance and supervision the writing of this thesis would have been impossible. Thanks are also due to Professor A. Adler for reading the thesis and for his valuable comments and suggestions.
The financial support of the University of British Columbia and the National Research Council of Canada is gratefully acknowledged.
I would also like to take this opportunity to express my thanks to Mrs. Y.S. Chia Choo for typing this thesis. Table of Contents
Page
Introduction
Chapter I Separation Axioms Between TQ and T
§1. Introduction 1
§2. Separation Axioms Between TD and T-^ 1
§3. Relations of the Axioms 6
Chapter II Properties of the Separation Axioms
§1. Introduction 10
§2. Strengthening of Topologies 10
§3. Product Spaces 16
Chapter III Minimal Topologies
§1. Introduction 21
§2. Minimal TQ and Minimal Tp Spaces 22
§3. Minimal Tx Spaces 28
§4. Minimal Regular Spaces 28
§5. Minimal Hausdorff Spaces 33
§6. A Characterization of Order Topologies by Minimal TD Topologies 36
Bibliography 43
\ Introduction
We shall say that a separation axiom implies another if every topological space which satisfies the first axiom also satisfies the second.
Separation axioms between Tc and T]_, that is, separation axioms which
imply TQ and are implied by T^, were first studied extensively by Aull and Thron [1]. They introduced a hierarchy of separation axioms between
TD and Tlf namely TUD, TD, T(Y), Tp, Ty, TYS, TDD and TFF, gave characterizations of them, and studied their properties. Later Robinson
m and Wu [10] defined T^ \ strong TQ and strong TD spaces. The first chapter of this thesis is devoted to a survey of these separation axioms.
Their relative positions are studied, and examples are given. We introduce a new axiom, namely a T^jlp space. At the end of the chapter we obtain a diagram which shows the positions of these axioms.
In the first part of chapter II we shall study the behaviours of the separation axioms under a strengthening of the topology, following the pattern of Park [9]. It will be found that with the exceptions of T(Y),
strong TQ and strong Tp, our axioms are preserved when the topology is
strengthened. Product spaces of a family of TD or (ma cardinal) spaces are considered in the second part of this chapter, as Robinson and
Wu did in [10]. The major result will be that it is not possible to define a separation axiom between T^ and T-^ which is preserved under arbitrary products.
Chapter III is devoted to the study of minimal topologies on a set. Given a set X, the family of all topologies defined on X is a - 2 -
complete lattice. We shall consider topologies in this lattice which are
minimal with respect to a certain topological property. Minimal TQ , minimal TQ, minimal Tp minimal T2 and minimal regular spaces are
considered. Characterizations of minimal TQ and minimal spaces are obtained by Larson [6], while that of minimal regular spaces is obtained by Berri and Sorgenfrey [3]. In the last section we produce a characterization
of order topologies on a set by means of minimal TQ topologies by Thron and Zimmerman [11].
Terminology and Notation
The terminology and notation used in this thesis follow those of
Kelley [5]. In chapter III, where results on filter bases are used, one may refer to Bourbaki [4],
A mention of the following terminologies is in order :
(1) A set is said to be degenerate iff it consists of at most one
element.
(2) For the closure of a point x, or more precisely, of the set
{x}, we shall write {x:}. For the associated derived set we
shall write {x}' . CHAPTER I
SEPARATION AXIOMS BETWEEN TQ AND T1
1. Introduction
In this chapter we shall describe various separation axioms
intermediate in strength between TD and T^. Emphasis will be given on those not included in [1], We shall describe the axioms, and for those not found in [1] examples and characterizations will be given. The reader is referred to [1] and [7] for examples and equivalent forms for separation axioms introduced in [1]. It will be observed that all the axioms can be described in terms of the behaviour of derived sets of points.
2. Separation Axioms Between TD and T^
To characterize separation axioms between TQ and T-^ it is convenient to introduce the concept of weak separation in a topological space.
Definition 2.1 A set A in a topological space (X,T) is said to be weakly separated from another set B iff there exists an open set G3A such that G H B =
A = {x} or B = {y}, we write x |— B and A I— y instead of
{x} |— B and A |— {y}> respectively.
The following axioms are introduced by Aull and Thron in [1] : - 2 -
Definition 2.2. A topological space (X,T) is called a
(a) TyD-space iff for every x e X, {x}' is the union of disjoint
closed sets ;
(b) Tp-space iff for every x e X, {x}1 is a closed set ;
(c) TpD-space iff it is TD and in addition for all x, y e X,
x ^ y, we have {x}' fl {y}' = <|>;
(d) Tp-space iff given any point x and any finite set F such
that x i F, either x |— F or F I— x ;
(e) Tpp-space iff given two arbitrary finite sets F-^ and F2 with
H F2 =
(f) Ty-space iff for all x, y e X, x ^ y, fx} (\ {y} is degenerate;
(g) Tyg-space iff for all x, y e X, x f y, fx) n (yl is either
<{> or {x} or {y} ; and
(h) T(Y)-space iff for every x e X {x}' is the union of disjoint
point closures.
The following three separation axioms are due to Robinson and
Wu [10] :
Definition 2.3. Let m be an infinite cardinal. A topological space
(X, T) is called a T^ space iff for every x e X, {x}=F fY( t\ {0±: i e I}) where F is closed, each 0^ is open, and card (I) = m. - 3 -
Definition 2.4. A topological space (X,T) is called a strong TD space iff for each x e X, {x}' is either empty or is a union of a finite family of non-empty closed sets, such that the intersection of this family is empty.
Definition 2.5. A topological space (X,T) is called a strong TQ space iff for each x e X, {x}' is either empty or a union of non-empty closed sets, such that the intersection of this family is empty and at least one of the non-empty members is compact.
In view of the fact that in a space {x}' =
and that in a TQ space {x}' is the union of closed sets, it is immediate
that strong TD and strong TQ lie between TQ and T-^ . The following theorem shows that this is also true for T^m^ spaces.
Theorem 2.1. The following are equivalent.:
(a) (X,T) is a T(m> space ;
(b) For each x e X, {x}' is the union of m closed sets , i.e.,
{x}1 = U {C^ : i £ 1} where each Cj_ is closed and card (I) =
= m ; and
(c) For each A C X such that card A <_ m, A' is the union of m
closed sets.
Proof (a) => (b) For x e X, if {x} = F n ( H {0± : i e 1} ) , then - 4 -
{x}» = {x} - {x}
= - (F n ( H {0± : i e 1} ))
= ( {x} - F) U ( U { {x} - 0± : i e I } )
= U { {x} - 0± : i e I }
since {x} C F, where card (I) = m. Hence {x}' is the union of m closed sets.
(b) => (c) Suppose A = {x^ : i E 1} where card I <_ m. Denote by C the set of w-limit points of A, i.e.,
C = {x E X : every neighborhood of x contains
infinitely many points of A} .
Then C is closed, and
A' = C U ( U { {x±}' : i e I } )
because A' contains the right hand side and if x E A' is not an w-limit point, then x E {x^}' for some x^ E A. Now by (b) each {x^}' is the union of m closed sets and since card I <_ m, A' is the union of m closed sets.
(c) => (b) obvious. - 5 - r
1 (b) =>(a) If {x} = U {F± : i e 1} where each F± is closed and card I = m, then
{x} = {3D - {x}'
= fx} - U {F± : i e 1}
= {x} n (X - F± : i e 1}
Hence (X,T) is a space.
v Corollary. (a) Each T '-space is a TQ space.
(b) Each Tp space is a space for any m.
(c) Each T_ space is a T^m^ space for some m.
A combination of and TK ' yields the following separation axiom.
Definition 2.6. Let m be an infinite cardinal. A topological space
(X,T) is called a T^m^ space iff for every x e X, {x}' is the union of m disjoint closed sets.
From the definition it is immediate that every space is a
T^UlP space for all m, and every T^j!j) space is a space as well
m as a T^ ) space. Moreover every TJJD space is a T^™^ space for some
cardinal m. Thus T^ lies between TQ and T1 . - 6 -
3. Relations of the Axioms
The following diagram is obtained by Aull and Thron in [1]. In
this diagram Ta —> Tg means that every Ta space is a Tg-space.
-> TD D -> TD
,YS
FF -> Ty > TF > T(Y) >'T1U D —> To
We shall now attempt to place the T^ and T^m^ spaces into this chart and show that they are new axioms intermediate in strength
between TQ and T-^ • By virtue of Corollary to Theorem 2.1 and the remark following Definition 2.6 we have
-> TD D -> Tr
T(m)
YS UD T(m)
-> T(r) > T FF -> Ty > Tp UD -> T,
We introduce some examples.
Example 1. X = real numbers.
closed sets :
Example 2. X = real numbers
closed sets :
of these sets.
Example 3. X = integers
closed sets : X and {n}, n / 0 plus finite unions of
these sets.
The reader is referred to [7] for detailed description of these examples.
Theorem 3.1. T^ and TUD are unrelated.
m Proof : (1) T^ ±> TUD : Example 1 is a T^ ) space where m = c,
but not a TyD space.
(2) TUD T^ : Example 2 is a TUD space but not a T^
space.
Theorem 3.2. T^ and T(Y) are unrelated.
Proof : (1) T(m) f> T(Y) : Theorem 3.1 Proof (1).
(2) T(Y) i> T^ : Example 2 is a T(Y) space but not a T(t^
space. - 8 -
For T^Q) we have the following theorem
m) Theorem 3.3. (a) TQ j» T^ .
(b) T
(c) T(Y) T
Proof : (a) Example 1 is a TQ space which is not a space for
any m.
(b) Example 2 is a space but not a T^ space.
(c) Example 2 is a T(Y) space which is not a space.
For strong Tp and strong TQ spaces we can make the following observations.
Theorem 3.4. (a) Every strong Tp space is a TD space.
(b) Every strong TD space is a T0 space.
Proof : This follows from the definitions.
Example 4. X = {a,b}.
T = U, {a}, X} .
This is a Tpp space, because {x}1 =
is closed for every x e X and for x f y, {x}' (\ {y}' =
it is not a strong TD nor a strong TQ space since {a}' = {b} cannot be expressed as the union of a family of non-empty closed sets whose intersection is empty.
We conclude this section by the following chart : CHAPTER II
PROPERTIES OF THE SEPARATION-AXIOMS
1. Introduction
In this chapter an attempt is made to investigate various properties of the separation axioms we have introduced. Properties like whether a separation axiom is preserved under strengthening of the topology
or under product are considered.
2. Strengthening of Topologies
It is known that the property of being TQ or T^ is preserved under strengthening of the topology. In this section we shall study the
axioms between TQ and T-^ . It will be found that the same is true for most of our axioms. The following lemma, which follows directly from the definition, will be useful. In this lemma and throughout the remainder of
this section, (X,T^) will denote a strengthening of a topological space
(X,T), where T and T-, are the two families of closed sets.
Let {x}, {x}' and {x}^ , {x}^ be the closures and
Lemma 2.1. (a) derived sets of the point x in T and respectively.
Then {X}1<1 {x} and {x}^ cZ {x}' .
Suppose = T U {Aa} where Aa is a closed set in (b) Tj_ for each a . Then if x I U Aa , then {x} = {x}p
{x}' = {x}[ . - 11 -
Proof : (a) is clear since we have more closed sets in than in T.
(b) since x i U Ag , the closed sets in containing x
are precisely those in T containing x, and the equality
follows.
Theorem 2.2. If (X,T) is a T^ space, then so is (X,-^),
Proof : Let x be an arbitrary point in X. We shall show that {x}^ is the union of disjoint closed sets in . Since (X,T) is T^ , {x}' =
= U Ca where Ca e T for each a and Ca H Ca, = <{> if a f a' . Now by Lemma 2.1 (a), {x}^ {x} , hence
{x}^ H {x}'
= {x}1 n ( u ca)
- u ( {x^ n ca)
where the last inclusion follows from the fact that each {x}^ H Ca is a
c = subset of {x}^ not containing x. Since ( Vx}^ n Ca) n ( {x}^ n ai)
<(> if a 4 a' ,{x}^ is the union of disjoint closed sets. Therefore
(X,^) is a TUD space.
Theorem 2.3. If (X,T) is a T^ space, so is (X,T^) - 12 -
Proof : We have seen in the Proof of Theorem 2.2. that the equality
{x}^ = .{1E>1 n {x}'
holds for every x e X. Thus if (X,T) is TD, then {x}' is closed in
T and so {x}^ is closed in . Hence (X,T^) is also T^ .
Theorem 2.4. T(Y) is not preserved under the strengthening of the topology.
Proof : Let X = {0, 1, 2, 3,•••}
T = {$} U { {n, n+1, n+2, • •'•} : n = 0, 1, • •'•}
Then (X,T) is a T(Y) space because for each n,
{n}' = {n+1, n+2, •••}
Now let
T± = T U { {n, n+2, n+4, • •'•} : n = 0,2, • •'•}
Then (X,!^) is a strengthening of (X,T) but (X,^) is not a T(Y) space since for each n
{2n - 1}[ = {2n - 1} (\ {2n - 1}'
= {2n - 1, 2n, 2n + 1,•••} (\ {2n, 2n + 1,•••}
= {2n, 2n + !,•••} - 13 -
cannot be written as a union of disjoint point closures.
Theorem 2.5. If (X,T) is a Tp space, so is (X.T^).
Proof : A space is a TF space iff for every x e X, y e {x}* implies
{y}' = <(.. (Cf. [1] Theorem 3.2). Now if y e {x}^ , then y E {X}1 and
hence {y}1 =
Theorem 2.6. If (X,T) is a ' Ty space, so is (X,^).
Proof : We have, for x, y E X, x f y,
n {y)1c {x} n {y} .
Since (X,T) is Ty , {x} n {y} is degenerate, so that {x} (\ {y^
is also degenerate.
Theorem 2.7. If (X,T) is a Tyg space, so is (X,^).
Proof : For x, y E X, x ^ y, since {x} f\ {y} is either <)>, {x} , or {y} , the same holds for fx}^ H ^^1 '
Theorem 2.8. If (X,T) is a TpF space, so is (X,T^).
Proof : (X,T) is TpF iff {x}* = <(> for all but at most one x E X
(cf. [1] Theorem 3.3). Since {x},' C (x}' , the same is true for {x}' . - 14 -
Theorem 2.9. If (X,T) is a TDD space, so is (X,^).
Proof : Since (X,T) is TD , by Theorem 2.3 (X.Tp is also TD
Now for x, y e X, x ^ y , {x}' (\ {y}' = <|>, hence {x}^ fV {y}^ = (j>.
Thus (X,^) is also TDD .
Theorem 2.10. If (X,T) is a T^ space, so is (X.T^)
Proof : Let x e X. Then
{x}' = U {C± : i e 1} where each e T and card I = m. But
{x}| = {3E>1 n {x}'
= H ( U C±)
= u ( n c±)
where {x}]^ n C± e Tj_ . Hence (X,^) is also T^,
Theorem 2.11. If (X,T) is a T^ space, so is (X,^)
Proof : Since (X,T) is T^m\ for each x e X
1 {x} = U C± , i e I - 15 -
where each £ T, card I = m and Cj H Ck =
{*}[ = {x}x H {x}'
= n ( u c±)
= u ( rax n Cj,)
where the family { {x}-^ H Cjj^} is disjoint. Hence (X,^) is a T^ space.
Theorem 2.12. Strong Tp is not preserved under the strengthening of the topology.
Proof : Consider the example :
X = {a, b, c}
T = U, {a}, {a,b}, {a,c}, X} . Here
{a}' = {b,c} = {b} U {c}
{b}« = (J> {c}' =
Hence (X,T) is a strong Tjj space. Now let T-y = {<{>, {a}, {b}, {a,b},
{a,c}, X}. Then (X,T^) is not a strong Tp space because {a}| = {c}.
Theorem 2.13. Strong TQ is not preserved under the strengthening of the topology. - 16 -
Proof : In the example in the proof of Theorem 2.12, (X,T) is a strong
TQ space but (X,T^) is not.
3. Product Spaces
An interesting question concerning the separation axioms is whether they are preserved under arbitrary products. The product space
of a family of TQ (or Tj) spaces is again TQ (or T-^. In this (m) section we shall consider products of TJJ and T spaces, m a cardinal.
Theorem 3.1. Let { (.X±,T/) : i = 1, 2, • • • ,n} be a finite family of TD n spaces. Then II X. is also a Tjj space. i=l
Proof : A space (X,T) is a TD space iff for every x e X, {x} = GHC, where G is an open set and C is a closed set in X (cf. [1] Theorem 3.1). n
, Now take x = {x^, x2, • *,xn} e II X^ . Then each {x^} = G.^ H , where
G^ and are open and closed respectively in X^ . Hence
N -1
{x} = n P. (G± n c.j) i=l
N -1 -1
= n (P/(G.) n P/(c.)) i=l
N -1 N -1 = n p. (G.) n n p/cc.) i=l 1 i=l 1 - 17 -
Thus {x} is the intersection of an open set with a closed set. Hence n
II X. is also TD . i=l
The following theorem shows that Tp is not preserved under arbitrary products.
Theorem 3.2. Let {X^ : i e 1} be an infinite family of TQ spaces,
where each X^ is not T-^ . Then II X^ is not a TD space. iel
Proof : For every iel, since X^ is not T-^ , there is a point otjL e X^ such that {a^}1 ^
Y. We shall show that {a}^ is not closed in Y. To this end we first observe that if x e Y - {a} and 0 is a basic open neighborhood of x, then p^(x) e {cT^} implies that e p^(0) for every i. Hence x e {a}y . Thus we have {a}^ = Y - {a} . It therefore suffices to show that Y - {a} is not closed (in Y). We observe that if 0 is a basic open neighborhood of a in Y, then
0 = Y n TKC^ : i e 1} where each (L is an open neighborhood of e X^ and 0^ f X^ for only a finite number of iel. Since I is infinite, there exists a non-empty subset I' d I such that 0^ = X^ for all iel' . Since we have {of.} - {a} ^
as follows
a± if i e I - I*
1 {a±} - {a} if iel
Then 3 = {$^ : i e 1} e 0. This means that every open neighborhood of a in Y contains points of Y - {a} . Thus Y - {a} is not closed in
Y.
We therefore have proved that the subspace Y of II X. is iel
not a TJJ space. Since the property of being TD is hereditary, II X^ iel is not a Tp space.
Theorem 3.3. Let (X,T) be a Tp space. Then all powers (that is,
products of X by itself) of X are TD spaces iff (X,T) is T^ .
Proof : This follows from Theorem 3.2.
Theorem 3.4. A product of m spaces is again a space.
Proof : Let II X^ be a product of the x^m^ spaces X^ , where iel card (I) = m. Let x = {x. : i e 1} be an arbitrary point in II X. iel
Then for every iel, {x.} = G. m (\ F. where F. is closed in X. X X y III 1 X X and x y_ i n is an intersection of m open sets in X.x . Thus - 19 -
{x} = H {p"1^) : i e 1}
-.n'^I1 1 = H (p^CGi^) : i e 1} H H {p~ (Fi) : i e 1} Now by the continuity of p^ , each p^(G^ m) is the intersection of m open sets in irX^ , and also H (p^CF^) : i e 1} is closed in irX^ . Since card (I) = m, we have expressed {x} as the intersection of a (m) closed set with m open sets. Thus TTX^ is T . Theorem 3.5. Let {X^ : i E 1} be a family of spaces none of which is T-^ , and let card (I) = n. Then T\X± is a T^ space iff n < m. (m) Proof : .If n <_ m, then by Theorem 3.4, irX^ is again a x space. If n > m, then by an analogous argument as in that used in the fro) proof of Theorem 3.2, TTX^ is not a T space. Theorem ,3.6. There does not exist a separation axiom between ancj T-^ which is inherited by arbitrary products. Proof : Let Ta be a separation axiom between and T-^ . Fix an infinite cardinal m, and let {X^ : i E 1} be a family of Ta spaces, none of which is T^ and suppose card (I) = n > m. Now each X^ is a - 20 - T space, so by Theorem 3.5 irX^ cannot be a T space since n > m. Therefore irX. is not a T„ space. CHAPTER III Minimal Topologies 1. Introduction Given a set X, the family of all topologies defined on X is a complete lattice. Of great interest are topologies which are minimal in this lattice with respect to a certain topological property, in the sense of the following definition. Definition 1.1. Let P be a topological property. A topology T defined on a set X is called a minimal P space iff T has property P and every strictly weaker topology on X does not have property P. Thus if P stands for TQ, TD, T-^, T2, regular, completely regular, normal or locally compact space, the topology T will be called a minimal TQ , minimal TD , minimal T2 , minimal regular, minimal completely regular, minimal normal or minimal locally compact topology accordingly. It is the purpose of this chapter to investigate some of these minimal topologies, obtain their characterizations and arrive at some of their properties. - 22 - 2. Minimal TQ and Minimal Tp Spaces For the characterizations of minimal T0 and minimal TD spaces the following Lemmas will be useful. Lemma 2.1. Let (X,T) be a TQ (Tp) space and let B be an open subset of X. Let T(B) = {G e T : GCB or B c G} . Then (X,T(B)) is a TQ (TD) space. Proof : We first show that T(B) is indeed a topology. Since and B d X, we have <|>, X E T(B). Now if G-j_, G2 e T(B), then either Be G^, B C. ^n wn^cn case B C H G2, or B contains one of the two sets in which case G-^ fl G2 C B . Hence G-^ H G2 e T(B). Finally if Ga e T(B) for every a e A, then either every GaC B or B cGa for some a, so that U Ga C. B or B C U GQ, whence U Gae T(B). aeA aeA aeA Now suppose T is a TQ topology. To show that T(B) is also TQ , let x, y e X, x ^ y and suppose that GeT, x£G,y^G. We consider three cases. (1) If x e B and y t B, then B is an open set in T(B) containing x but not y. Similarly if y e B and x $. B. - 23 - (2) If x, y e B, then G H B is an open set in T(B) containing x but not y. (3) If x, y i B, then G U B is an open set in T(B) containing x but not y. Thus (X,T(B)) is also a TQ space. Suppose next (X,T) is a T^ space. TQ prove that T(B) is also TD, take arbitrary x e X, and consider , the derived set of x in the topology T(B). Again we consider two cases. (i) If x i B, then x e X - B which is closed in T(B). Thus {x}lCL X - B and so BC X - {x}' . Hence X - {x}' is open, and so D a a {x}' is closed in T(B). a (ii) If x E B, we can prove that if y e B, y $ x, then y $. {x}' . B Indeed, in this case B - {x} e T(B), y e B - {x}, but x I B - {x} . Hence {x}' (\ B = d> , which means that BCX - {x}' . Hence {x}' is closed in T(B). Lemma 2.2. In a topological space (X,T), the following are equivalent: (1) The open sets in the topology are rested, i.e., for A, B e T, either A C B or B CA. (2) The closed sets in the topology are rested. (3) Finite unions of point closures are point closures. - 24 - Proof : The equivalence of (1) and (2) is clear. That (2) implies (3) is also obvious, since the union of a finite number of point closures is the largest one. To show that (3) implies (2), let C, D be two non-empty closed sets in (X,T), and suppose C ^ D. Then either C - D Hence {x} = {z} or {y} = fz}. But {y} = {~z} is impossible since this would imply x e {¥} = {y} CD, a contradiction because x i D. Therefore fx} = {z} and so y e {z} = fx} C C. We have thus proved that DC C. Similarly if D - C f The following Theorem gives a characterization of minimal TQ spaces. Theorem 2.3. A TQ topological space (X,T) is minimal TQ iff the family g = {X - fx} : x e X} U {x} is a base for T and finite unions of point closures are point closures. Proof : ( >) Let A, B be open sets in T. If A^B, B cj/t A, then by Lemma 2.1, T(B) is a TQ topology on X such that T(B) cT and A £ T(B). This contradicts the minimality of T. Thus either A CB or B C A. Hence by Lemma 2.2, finite unions of point closures are point closures. Now since T is a nested family of open sets, the - 25 - subfamily {X - {x} : x e X} U {X} is closed under finite intersections, and so is a base for some topology on X. Clearly T^CT. Also T is a TQ topology, because for x ^ y, either x i {y} or y i {xl since T is TQ . By the minimality of T, we have = T. So 3 is a base for T. ( base for T. Suppose T*d T, where T* is a TQ topology. Suppose there is a point x e X such that {x} ^ {x:}* , where fx}* is the closure of {x} in T* . Since Vx} {y} C {y}* , hence x e {x} C {y}* . Thus x e {y}* and y e {x}* , which is impossible in a TQ space. Hence for every x e X, {x} = {xT}* . Since 3 = (X - {x} : x e X} U {X} is a base for T, we must have T = T* . Thus T is minimal TQ . Theorem 2.4. A TD topological space (X,T) is minimal TD iff finite unions of point closures are point closures. Proof : ( y) By a proof identical to the one in Theorem 2.3. (<—) Suppose (X,T) is TD and T is nested. Let T*C T, where T* is a TD space. We shall show that T* = T. Since a TD space is a TQ space, we can apply the argument in Theorem 2.3 and arrive at the conclusion that fx} = fx}* , {x}1 = {x}'* for every x e X. Now assume that T* ^ T. Then there is a set CCX closed in T but not - 26 - closed in T* . If C* denotes the closure of C in T*, then CCc*. ft; But T* is TQ , therefore C* - C consists of exactly one point x. Since there does not exist a closed set in T* smaller than C* which contains x, we have C* = {x}* = C U {x} = {x}'* U {x} . Thus it follows that C = {x}'* because x ^ C, x^{x}'*. But this contradicts the fact that C is not closed in T* . Thus we must have T* = T and T is minimal T^ . The following two examples show that the two conditions in Theorem 2.3 cannot be relaxed and that they are independent of each other. Example 2.1. X = real numbers T = {(-°°,x) : x e X} U {(-°°,x] : x e X} U { (X,T) is a TQ space in which the open sets are nested, but is not a 1 minimal TQ space because, for example, the proper subfamily T = = {(-°°,x) : x e X} U { Example 2.2. X = {a,b,c} T = {<)>, {a}, {b}, {a,b}, X} . (X,T) is a TQ space. Moreover, since {¥} = {a,c}, {b} = {b,c}, {c} = {c}, the complements of these sets are {b}, {a}, {a,b} which form a base for the topology T . However, the open sets are obviously not nested. - 27 - The next example shows that minimal TQ is not hereditary. Example 2.3. X = real numbers. T = {(-°°,x) : x e X} U {<}>,X} A = (-00,0] U (I,-) (X,T) is a minimal TD space since the open sets are clearly nested and for every x e X, fx} = [x,«>) so that the family {X - fx} : x e X} is precisely T itself. However, the subspace A is not minimal TQ because although the open sets are again nested, the complements of point closures do not form a base. Indeed, we first observe that (-°°,0] is open in A and 0 e (-°°,0] . Now if x e (-°°,0] , then 0 e [x,<») (\ A = = fx} H A = fx}^ and so 0 i A - {x}^ • 0n tne other hand, if x e (l,00), then A - fx}^<^: (-co»0] • Hence {A - fx}^ : x e A} is not a base for the relative topology. For minimal T^ spaces we have the following result. Theorem 2.5. Every subspace of a minimal Tp space is again minimal Tp. Proof : Each subspace of a TQ space is TQ. By the definition of relative topology the nestedness of open sets is inherited. Hence by Theorem 2.4 and Lemma 2.2 the result follows. - 28 - 3. Minimal T-^ spaces For minimal T-^ space we have the following neat theorem. Theorem 3.1. A topological space (X,T) is minimal T-^ iff the non- trivial closed sets are precisely the finite sets. Proof : Given any set X let T* = {A C X : X - A is finite} U {<(>} . Then it is well-known that (X,T*) is a T-^ space. Moreover, T* is the weakest T-^ topology on X because if T is another T-^ topology on X, then all finite sets are closed in T and so T* Corollary 3.2. Any subspace of a minimal T-^ space is minimal T-^ . 4. Minimal Regular Spaces For subsequent discussion we shall make use of the notion of a filter base. The reader is referred to [4] for definitions and results concerning filter bases in a topological space. For our present arguement we introduce some definitions. - 29 - Definition 4.1. A filter base F on a set X is said to be weaker than a filter base G on X iff for each F e F, there exists some G e G such that G C F. Definition 4.2. A filter base F on a set X is said to be equivalent to a filter base G on X iff F is weaker than G and G is weaker than F. It is readily checked that the relation of equivalence between filters is an equivalence relation. Definition 4.3. A filter base F on a topological space (X,F) is called an open (closed) filter base iff for every F e F, F is an open (closed) set. Definition 4.4. A filter base T on a topological space is called a regular filter base iff it is open and is equivalent to a closed filter base. Definition 4.4 is suggested by the following theorem. Theorem 4.1. In a regular topological space (X,T), the filter base qf open neighborhoods of a point is regular. Proof : Let B(x) be the filter base of open neighborhoods of the point x and let C(x) be the filter base of closed neighborhoods of x. - 30 - Obviously for every C e C(x), there is a Be B(x) such that Be C Since T is regular, for every B e B(x), we can also find C e C(x) such that C C B. Hence B(x) is equivalent to C(x) and so is regular. We shall be interested in the following conditions in a topological space : (a) Every regular filter base which has a unique cluster point is convergent to this point. (3) Every regular filter base has a cluster point. Theorem 4.2. A regular space (X,T) is minimal regular iff (a) holds, Proof Necessity : Suppose $ is a regular filter base which has the unique cluster point p, and assume that 3 does not converge to p. We shall construct a topology on X which is regular but strictly weaker than T. For each x e X, let U(x) be the filter base of open neighbor• hoods of x. Define U(x) if x i p U'(x) = {U U B : U e U(p), B e 8} if x = p Under this definition there is defined on X a topology T' with U'(x) as an open neighborhood base at each x e X. Now since B does not converge to p, there is a U e U(p) - B which does not contain any set - 31 - in U'(p). Hence T* is strictly weaker than T. To show that T' is regular, first it is clear that T' is regular at each point x / p. At the point p, since B is regular, B is equivalent to some closed filter base C. Now if p e U U B, where U E U(x), Be S, there exists closed sets V, C such that p e VcTlU, C e C, C C B, so that p e VUC Thus T' is also regular at p. This shows that T is not a minimal regular topology. Sufficiency : Suppose (X,T) is regular and satisfies (a). Let T1 be a regular topology on X such that T'C T. For each x £ X, denote by U(x) and K'(x) the open neighborhood systems of x in T and T1 respectively. Since T' is regular, the filter base U'(x) is, by Theorem 4.1, T1-regular. Moreover, x is the only cluster point of U'(x). Since T'C T, it follows that U1 (x) is regular in T and has x as its unique cluster point. By (a), U'(x) must converge to x in T, so that by definition (J(x)CU'(x). But T'C_T implies U'(x)CU(x). Thus U (x) = Li' (x) and we have T' = T. Hence T is minimal regular. Lemma 4.3. If the subspace A of the regular space (X,T) satisfies (g), then A is closed in X. Proof : Suppose A ^ A and let p e A - A . Let U and 1/ be the open and closed neighborhood systems of p in X, respectively ( V is a neighborhood system since T is regular). Let - 32 - B = {A H U : U e 0} C = ' {A n V : V e V} Then B is an open filter base in A, while C is a closed filter base in A. Moreover, since T is regular, U is equivalent to f, and hence B is equivalent to C. Thus B is a regular filter base on A. But B is also a filter base on X, and U is weaker than B. Since p is the only cluster point of U in X, it is also the only cluster point of 8 in X. This means that, since p i X, g has no cluster point in X, which contradicts (g). Hence A must be closed. Theorem 4.4. In a regular space, (a) implies (g). Proof : Let B be a regular filter base on a regular space (X,T). Assume B has no cluster point. Let C be a closed filter base equivalent to B. Fix p e X. Let U, V be the open and closed neighborhood systems of p respectively. Then U and V are equivalent, by Theorem 4.1. Let F = {B U U : B e 8, U e U} and G = {C U V : C e C, V e I/}. Then F is an open filter base, and G is a closed filter base on X. Moreover, F is equivalent to G, which follows from the equivalences of B to C and U to (/. Thus F is regular. Now p is a cluster point of F, and since B has no cluster point, F has no cluster point other than p. However, F does not converge to p. This contradiction to (a) shows that (a) implies (g). - 33 - Theorem 4.5. A minimal regular subspace of a regular space is closed. Proof : By Theorem 4.2, a minimal regular subspace of a topological space satisfies (a). By Theorem 4.4, it also satisfies (g), so that by virtue of Lemma 4.3 it is closed. Corollary 4.6. Minimal regularity is not hereditary. 5. Minimal Hausdorff Spaces For the characterization of minimal Hausdorff spaces we consider the following two properties of a topological space : (1) Every open filter base has a cluster point. (2) Every open filter base which has a unique cluster point converges to this point. Theorem 5.1. In a Hausdorff space, (2) implies (1). Proof : Suppose (1) does not hold, and let B be an open filter base which has no cluster point. Fix p e X. Let U be the open neighborhood system of p. Define G = {V U B | V e U and B e B} - 34 - Then G is an open filter base and p is its only cluster point. By (2), G converges to p .'"But 8 is weaker than B, so B also converges to p. But this contradicts the assumption that 8 has no cluster point. Hence (1) holds. Theorem 5.2. A Hausdorff space (X,T) is minimal Hausdorff iff (2) holds in T. Proof : Necessity : Let (X,T) be Hausdorff and suppose that (2) does not hold, so that there exists an open filter base 8 having the unique cluster point p but 8 does not converge to p. For each x e X, let U(x) denote the open neighborhood system of x. Define for every x a family of subsets of X as follows : U'(x) = U(x) if x ^ p, and U'(p) = {U U B : U e U(p), B e B}. With this definition there exists a topology T1 on X with U'(x) as an open base at each x. That T'O T is clear. Moreover since 8 does not converge to p, there is U e U(p) - 8 and U does not contain any set in U'(p). Thus Tf<~T. We now show that T' is Hausdorff. Indeed, for x, y e X, x ^ y, if x ^ P> y ^ P» then the existence of disjoint open neighborhoods is guaranteed. For x f p, by Hausdorffness of T there are A e U(p), D e U(x) such that A H D = <|>. Since p is the only cluster point of 8, x cannot be a cluster point and so there is E e U(x), B E 8 such that E H B = (A u B) H (D n E) = <\>. Thus T' is Hausdorff and T is not minimal Hausdorff. - 35 - Sufficiency : Let (X,T) be Hausdorff satisfying (2) and let T' be a Hausdorff topology on X with T'C T. Let x e X, and let U(x) and U'(x) be the open neighborhood systems of x in T and T' respectively. Then U'(x)c~. M(x). The open filter base U'(x) has x as its only cluster point, because T' is Hausdorff. Since T'<^ T, x is the only cluster point of U'(x) in (X,T). By (2), U'(x) converges to x. Thus U(x) Theorem 5.3. Let X be a subspace of the Hausdorff space (Y,T). If X satisfies (1), then X is closed in Y. Proof : If X X , let p e X - X . Let U be the open neighborhood system of p in Y. Then B = {X H U : U e U} is an open filter base on X. Moreover, as a filter base on Y, B is stronger than U and since p is the only cluster point of U, B cannot have any other cluster point than p in Y. This means that B has no cluster point in X, since p i X. But this contradicts the hypothesis that X satisfies (1). That the property of being minimal Hausdorff is not hereditary is shown by the following theorem. Theorem 5.4. A minimal Hausdorff subspace X of a Hausdorff space (Y,T) is closed. - 36 - Proof : Since X is minimal Hausdorff, Theorems 5.1 and 5.2 tell us that X satisfies property (1). Theorem 5.3 then concludes that X is closed. Theorem 5.5. If a subspace of a minimal Hausdorff space is both open and closed, then it is minimal Hausdorff. Proof : Let A be an open and closed subset of the minimal Hausdorff space (X,T). Let B be an open filter base on A with only one cluster point p e A. Since A is open in X, B is also an open filter base on X. Since A is also closed, the closure of B e B in A is closed in X, and hence p is the only cluster point of B on X. But now X is minimal Hausdorff, hence B converges to p on X, by Theorem 5.2. Since p e A, B also converges to p on A. Again invoking Theorem 5.2, A is minimal Hausdorff. 6. A Characterization of Order Topologies by Minimal TQ Topologies In this section we shall give a characterization of order topologies on a set X by means of minimal TQ topologies on X. We recall that a topology T on X is an order topology on X iff there exists a linear order <_ on X such that the sets of the forms {y : y < x} and {y : x < y}, where x e X, form a subbase for T, where a < b means that a <_ b but a ^ b. We prove that a topology T on a set X - 37 - is an order topology iff (X,T) is T-^ and T is the least upper bound of two minimal topologies on X, in the sense of the following definition. Definition 6.1. A topology T on a set X is the least upper bound of two topologies Ty and T"2 on X iff T is the smallest topology containing 7"-^ and T"2 . We shall write T = V T2 in this case. Lemma 6.1. If and T2 are topologies on X and and B2 are bases for and 7"2 respectively, then B-^ u B2 is a subbase for T± V T2 . Proof : It is clear that B-^ U B2 is a subbase for some topology, say T, on X. Also T^C. T and T2C T . Hence V T2 C T . On the other hand any topology on X containing and T2 must contain B-^ U B2 and hence contains unions of finite intersections of members of B^ U B2, so that it contains T. Thus TcT^ V T2 • Definition 6.2. Let T be a topology on a set X. Define a relation £ j on X as follows a <_ j b iff b e {a} It is immediate that <^ j as defined above is reflexive and transitive. In a TQ space it is also anti-symmetric, as the following lemma shows. - 38 - Lemma 6.2. A topology T on a set X is TQ iff <_ j is a partial order. Proof : If T is TQ and a< yb, b <_j a, then b e fa} and a e {b}, so that a = b since T is TQ . Conversely, if <_ j is anti- symmetric, and if a ^ b and b e {a} , then a i {b} . Hence T is TD. Lemma 6.3. The topology T is nested iff any two elements are comparable, i.e., for a, b e X , either a <_ j b or b <_ j a . Proof : If T is nested, then for a, b e X, either {ID C {b} or {b} C {a} , and so either aji-j-b or b <_ -j- a . Conversely, suppose the condition holds. Let A, B e T and assume A cji B. Choose a e A - B . Now for each x e B, x i fa} since B is an open set containing x but not a. Thus a {_ j x and so x <_ j a. Therefore a e {xl and since a e A e T , we must have x e A. Hence B C. A and so T is nested. Theorem 6.4. A topological space (X,T) is minimal TQ iff <_ j is a linear order and {{y : y < j x} : x e X} U {X} is a base for T. Proof : If T is minimal TQ ,> then by Theorem 2.3 and Lemma 2.2 T is nested and {X - fx} : x e X} U {X} is a base for T. It follows from Lemma 6.2 and Lemma 6.3 that <_ j is a linear order. Moreover for each x e X, - 39 - X - {x} = X - {y : y e' {x}} = X - {y : x <_ j y} = {y : y < j x} since <_ j is linear. Thus {{y : y In the other direction suppose <_ j is linear and {{y : y < x} : : x e X} U {X} is a base for T. Then by Lemma 6.3, T is nested. Again, because <_j is linear we have for each x e X, X - {JC} = {y :- y < j x} . Thus {X - {x} : x e X} U {x} is a base for T and by Theorem 2.3 T is a minimal TQ topology. Theorem 6.5. Given any set X, there exists a 1-1 conespondence between the set of all minimal TQ topologies on X and the set of all linear orders on X. Proof : Let M be the set of all minimal topologies on X, and let L be the set of all linear orders on X. Define {{y : y < x} : x e X} U {X} and {{y : y < _ x} : x e X} U {X} , 'l '2 respectively. Since the two basis are the same, = T2 • - 40 - Now let < be a linear order on X. Let B = {{y : y < x) : : x e X} U {X} . For any x, z e X, {y : y < x} (\ {y : y < z} is either {y : y < x} or {y : y < z}, since <_ is linear. Thus B is a base for some topology on X, say T. Let <_ j be the relation on X defined by T. We shall show that <_ y = <_ . If a <_ b and b e N e T then there is a B e B such that beBdN. If B = X then a e B C N. If B ± X then B = {y : y < c} for some c £ X. Then since a <^ b and b < c we have a < c and so a e BC N. This means that every open set containing b contains a. Hence b e {a} and a j b. On the other hand, if a <^ j b but b < a, then we have b e {y : y < a} e T but a i {y : y < a}, so that b i (a} and this contradicts a <_ j b . Hence a <_ b since < is a linear order. We have thus prove that <^ j = <^ . Hence by Theorem 6.4 T is a minimal topology on X. Consequently The following theorem gives the main result of this section. Theorem 6.6. A topology T on a set X is an order topology iff T is T^ and T is the least upper bound of two minimal TQ topologies. Proof : Let T be an order topology and let <^ be the associated linear order. Then the sets of the forms {y : y < x} and {y : x < y} form a subbase for T. Clearly T is T-^ . Let = {{y : y < x} : x e X} U {X} - 41 - AND B 9 = {{y : y > x} : x e X} u {X} . are As in the proof of Theorem 6.5, B-^ and B2 bases for topologies Ty and T2 respectively which are minimal TQ on X, and we have <_ = <^ and <_ = <_""'" where a <_^~ b iff b <_ a. By Lemma 6.1 71 ' 2 ~ " B-^ U B2 is a subbase for V T2 . But as mentioned above B^ u B2 is also a subbase for T. Hence T = V T2 is the least upper bound of two minimal TQ topologies. Conversely, assume that T = T^ \/ T2 where and 7"2 are minimal TQ and that T is T-^. Then we know that <_T and <_T are ' 1 ' 2 linear. We shall show that <^ =f.-r »i«e., a 1. T b iff ^ ^_ -r a . !l 2 '.1 '.2 a For this purpose suppose a <_T b, £. T b and a ^ b, and let G e T '.1 2 = V T2 be such that b e G. Then there exist G-^ e and G2 e T2 a such that b e G-, H G2 C G. Since £r b, b e {a}-r and so a E G-^ . 1 1 Similarly a e G2 . Thus a e G. This means that every open set in T containing b also contains a, which is impossible since T is T-^ . Hence for a, b e X, a / b and a <^T b, we have a j_ j b and since 'l 2 <_ is linear, we must have b <_ y a . Similarly b <^ j a, b / a ~'.2 2 2 implies a < _. b. Hence < T = < ^ . Now since T. and T„ are ~ 'l ~ !l ~ 2 minimal TQ , by Theorem 6.4 they have bases B-L = {{y : y < T x} : x E X} U {X} - 42 - and B9 = {{y : y > y x} : x e X} U {X} 1 1 respectively. By Lemma 6.1, B^ U B2 is a subbase for '/ T2 = T . But since ^ is linear, 8^ U B2 is the subbase for an order topology on X, which in this case must therefore be T. Thus T is an order topology. BIBLIOGRAPHY 1. Aull, C.E. and Thron, W.J., "Separation axioms between TQ and T^", Indag. Math. 24 (1962), 26-37. 2. Berri, M.P., "Minimal topological spaces", Trans. Amer. Math. Soc. 108 (1963), 97-105. 3. Berri, M.P. and Sorgenfrey, R.H., "Minimal regular spaces", Proc. Amer. Math. Soc. 14 (1963), 454-458. 4. Bourbaki, N., "Elements of mathematics general topology", Addison- Wesley Publishing Co., Inc., Mass. U.S.A. 5. 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