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On Hereditary Properties of Topological Spaces 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 <LX A the complement of the set A relative to > th e s e t f | z the restriction of f to Z where Z c x and a function defined on X Capital Latin letters such as A usually denote sets, while capital 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 <ic, the complement of C, is open in X. That is, d e e p . 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.
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