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Some Properties of Certain Subsets of Infinite Dimensional Spaces. William Philip Barit Louisiana State University and Agricultural & Mechanical College

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Some Properties of Certain Subsets of Infinite Dimensional Spaces. William Philip Barit Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1971 Some Properties of Certain Subsets of Infinite Dimensional Spaces. William Philip Barit Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Barit, William Philip, "Some Properties of Certain Subsets of Infinite Dimensional Spaces." (1971). LSU Historical Dissertations and Theses. 2023. https://digitalcommons.lsu.edu/gradschool_disstheses/2023 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. 72-3459 BARIT, William Philip, 1945- SOME PROPERTIES OF CERTAIN SUBSETS OF INFINITE DIMENSIONAL SPACES. The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1971 Mathematics University Microfilms. A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED SOME PROPERTIES OF CERTAIN SUBSETS OF INFINITE DIMENSIONAL SPACES A dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by William Philip Barit B.S.t Wayne State University, 1967 August, 1971 PLEASE NOTE: Some Pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS ACKNOWLEDGEMENT The author wishes to thank Professor R.D. Anderson for making it possible to travel with him to Amsterdam on his sabbatical, and for his advice and encouragement while directing this dissertation. ii TABLE OF CONTENTS ACKNOWLEDGEMENT. ................................. ...... ■ ABSTRACT................................................... CHAPTER I: TERMINOLOGY AND DEFINITIONS............. ..... CHAPTER II: PRELIMINARY RESULTS AND OBSERVATIONS - . CHAPTER III: AN c-HOMEOMORPHISM EXTENSION THEOREM........ CHAPTER IV: OTHER COMPACTIFICATIONS OF s AND RELATED SPACES CHAPTER V: A DISCUSSION OF R ' ( 1 )......................... - CHAPTER VI: OTHER SPACES RELATED TO R (n).-................ CHAPTER VII: CONTRACTION OF SPACES OF HOMEOMORPHISMS.... BIBLIOGRAPHY...................................... ....... VITA ABSTRACT In this dissertation ve investigate some properties of certain subsets of infinite dimensional spaces. Results in three general areas are obtained. The first area concerns the extension of homeomorphisms between special subsets of Hilbert space. An £ -homeomorphisms extension theorem is proved which shows that small homeomorphisms between these subsets can be extended to space homeomorphisms. Results in the second area concern properties which most of the commonly studied infinite dimensional spaces do not have, These properties include having points which are not negligible, being non-homogeneousf and having no non-degenerate factors. Certain remainders of compactifications of infinite dimensional spaces are shown to possess these properties. The last area concerns contractibility. Certain subsets of the space of homeomorphisms of some infinite dimensional spaces are shown to be contractible. CHAPTER I TERMINOLOGY AND DEFINITIONS In this dissertation certain infinite dimensional spaceB will be studied. This first chapter provides some necessary background and definitions. The symbolB X, Y, and Z will represent arbitrary topolo­ gical spaces unless otherwise noted. Taking the topological product of infinitely many suitable spaces is one of the most cosmon ways o£- producing an infinite dimensional space. Let X*” denote the space It. „ X. , where each X. is homeomorphic to X. In the case X is a i 0 i l n-dimensional manifold, the term basic neighborhood refers to a finite product of open n-cells cross the rest of the factors. The projection OD # map from X onto X^ is denoted by . A point x in X will be conmonly written (x^) or (x1 .x^x^, • • •) where x^ * tkx. If d* is a bounded metric on X, then the following metric is used for X . d(x,y) - LlL >q (2“1d ’(xi ,yi))2 ]^. Such metric spaces will be called infinite product spaces. Sometimes subsets of infinite product spaces are given different topologies than the subspace topology. In this case they will be called infinite coordinate spaces. The coordinate representation of points and notation for projection will be the same as for infinite product spaces. A subset K of either such space is said to be deficient in the i^*1 direction if it^K consists of a single point. K is said to be infinitely deficient if it is deficient in the i^*1 direction for infinitely many i. Let 0 be some specified point of X, then X* denotes {x £ X™! all but 1 2 - « finitely many coordinates are 0). Let 0 be the point in X with each coordinate equal to 0 . Let I be the closed interval [-1,1] and I be the open interval n n on (-1,1). The interior of the n-cell I (int I ) is I , and the boundary (3In ) is In\ int In . The n-1 sphere, Sn_1, is another name for 3ln . Let R be the real line. The following infinite product space has been widely studied, and certain subsets of it play an important role in infinite dimensional flO topology. The compact Bpace, I , is called the Hilbert cube and is customarily denoted by Q. The following subsets of Q are analogous to faces in finite dimensional cubes. WT ■ {x e Q I x. “± 1}. These are l 1 l th cabled the end faces in the (positive. negative respectively) i__ direction. Note that each endface is itself homeomorphic to Q. We call B(Q) “ > o ^Wi U Wi^ the PSeudo~boun<^ary Q* We mfty equivalently define B(Q) as (x e Q( at least one coordinate of x is t 1 L The complement of B(Q) in Q will be called the pseudo-interior of Q and is denoted by s . Note that s is the infinite product space 1 . For b belonging to (0,1), call C-b,bJ a standard subcube is s. Two other subsets of Q that have been useful are: £ ■ (x t sj all but finitely many coordinates of x belong to [-b,b3) and o ■ (x € s| all dc but finitely many coordinates of x are 0). Note o is I . Hilbert space. defined below, ia an infinite dimensional coordinate space which has played an important role in infinite dimen- sional topology. Let 1,, - (x e R | [. > ^ x. < (i.e. all square sumable sequences of real numbers)with the following metric. dU,y) “ > o (xi’yi)2]J* 3 Sometimes it is useful to regard Q and s as subspaces of 1^. It is easy to verify that the convex set {x e lgl |*^| is naturally isometric to Q. The following definitions deal with functions. Let H(Y) be the set of all homeomorphisms of Y onto Y. Let C(X,Y) be the set of all con­ tinuous functions from X into Y, and let E(X,Y) be the set of all em­ beddings of X into Y, If Y has a bounded metric d, we define the dis­ tance between f and g, p(f,g), as sup (d(f(x),g(x))|x in the domain of f and g>. The identity function from X onto X will be denoted by id , A or if no confusion is possible by id. If f belongs to C(X,Y) and A is a subset of X, then f |A represents the restriction of f to A. Also if g belongs to C(A,Y) and f|A * g, then we say f extends g. Let Ha (x ) ■ th € H{X)|hjA“ id}. Let f belong to C(X,Y) and g belong to C(X,Z), then (f,g) represents the function in C{X,Y ' Z) where (f,g)(x) ■ (f(x),g(x)). Let f belong to C(X,Y) and g belong to C(X',Y') then f * g represents the function in C(X * X'# Y * Y *) where (f * g)(x,x') » (f(x),g(x*)}. A map F in C(X * J,Y), where J is a closed interval, is called a homotopy. For each t in J, the map F in C(X,Y), where “ F(x,t), is called a level of the homotopy. If each level is a homeomorphism, F is called an isotopy. For U an open subset of X, h in H(X) is said to be supported in U if “ icl* Given (h^)^ > Q in H(X)f h in H(X) is called the —infinite — — ----left product of (h.)i and is denoted by LH.j. 0 _h.x if for each p in X, h(p) is the limit of the sequence thi ° hi_ 1 >... h 1 Cp)) a Q Let K be a subset of X. A homeomorphism, h, that is an infinite left product, LJTh^, is said to be locally finite mod K if the following is satisfied. u For each p in X\K, there is a neighborhood U of p and an in­ teger, N, such that for each n j N, hn|hN ^ . .^h1(U) “ ld* 00 If K ■ P, h is called locally finite. For h in H(X ) we use the following terminology. We say h is independent of the coordinate. if h ■ h* * id where h' belongs to H(IL..X.). The homeomorphism is simple \ lfK i if it is independent of all but a finite number of coordinates. The concept of locally finite mod K in the caBe of infinite product spaces is particularly useful when each h^ is simple. Then all points outside of K are subjected to motions which affect only a finite number of coordinates. Given and subsets of X, K 1 is said to be equivalent to K2 if there exists an h in H(X) such that h(K^ T w o points x and y are equivalent if {x} and {y} are.
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