Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 5-1973 Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies James A. Capps Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/gradreports Part of the Geometry and Topology Commons Recommended Citation Capps, James A., "Infinite Product Spaces Under the Tychonoff and Goofynoff Topologies" (1973). All Graduate Plan B and other Reports. 1137. https://digitalcommons.usu.edu/gradreports/1137 This Report is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Plan B and other Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. INFINITE PRODUCT SPACES UNDER THE TYCHONOFF AND GOOFYNOFF TOPOLOGIES James A.by Capps A report submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Plan B UTAH STATE UNIVERSITY Logan, Utah 1973 ACKNOWLEDGEMENTS I would like to express my thanks to all chose professors who have influenced the preparation of this report. I especially wish to thank my major professor, Dr. L. D. Loveland for his many valuable suggestions and criticisms and for the encouragement he offered me. I would like to thank Dr. Robert Heal for his suggestions and guidance. I wish to express my gratitude to my wife for her great patience which she has shown me during the writing of this report. James A. Capps TABLE OF CONTENTS INTRODUCTION ........ Page I. DEFINITIONS AND NOTATION 1 II. THE GOOFY CUBE .... 2 III. PRODUCTIVE PROPERTIES. 146 LITERATURE CITED 23 VITA ... 24 LIST OF TABLES Table Page w w 1. A Comparison of I and G 13 2. Productivity of Topological Properties 15-17 INTRODUCTION The only topology considered for the infinite product of topolo­ gical spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topo­ logy of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not re­ fer to any person of that name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of the popular topology texts provide much indication of why the Tychonoff topology is generally chosen instead of the more natural Goofynoff topology. This paper provides the reasons from a topologist's point of view. In Section III of this paper we compare the productivity of various properties under these two topologies. This paper was motivated by a question asked in the notes for a topology class at Utah State University in the fall of 1972. This question, concerned with the connectedness of an infinite product of connected spaces (under the Goofynoff topology), went unanswered through two graduate classes. An example showing that such a space may fail to be connected is given in Section II. In some respects the w example resembles the Hilbert Cube I and, for that reason, is called w the Goofy Cube and is denoted by G . In order to emphasize the difference, we include a table at the end of Section II which compares w w the topological properties of I with those of G . 2 I. DEFINITIONS AND NOTATION Given two sets A and B, the product, denoted AXB, is defined to be the set of all ordered pairs (a,b) where aEA and bEB. If X and Y are topological spaces, then the family of all sets of the form DXV, where U is open in X and Vis open in Y is a basis for a topology for XX Y. The topology generated by this basis is called the product topology for XXY. The definitions and remarks given thus far easily extend to products of any finite collection of topological spaces. An equivalent basis for the product topology on a finite product X1 X X2 X · • • X Xn is the set of all products of the form . U 1 X U2 X • · · X U0 where each Ui is restricted to a fixed basis for Xi This fact allows us to picture a basis element for the product topology as a n-sided box and thus many people refer to the product topology as the "box" topology. The mappings from XXY onto X and Y defined by TI ( (x,y) ) = x X and IT ( (x,y) ) = y, respectively, are called projections or Y projections onto the coordinate spaces. As can be easily shown, these projections are continuous and open. The box topology is the smallest topology on XXY which makes these projections continuous. We now extend these ideas on product spaces to infinite products. Let A be an indexing set (possibly uncountably infinite) and for each aEA, let A a be a topological space. The cartesian product a X {Aa: EA} (abbreviated X Aa where only one indexing set is involved) is the set of all functions f: A-UAa such that f (a) E Aa for each aEA. This product could also be thought of as a set of "infinite-tuples" 3 th where the a coordinate would be f(a), an element of Aa. For each 13 E: A, the map IT : X Aa....._ A , defined by I1 (f) = f (8), is called the 13 8 8 l3th th projection mapping of X Aa on to A . A is called the 8 factor 8 B th space or 8 coordinate space. We denote an element f of X Aa by th <xa> or more frequen tly by <f(a)> and call f(a) the a coordinate of f. There are many ways to describe a topology on the cartesian pro­ duct X Aa. Of course we shall want a topology which agrees with the box topology in the case of finite products and we would hope that the projection mappings would be continuous. The most well-known topology for X Aa, called the Tychonoff topology, is defined as the smallest topology for which the projection mappings are all continuous. The family of all subse ts of X Aa of the form X Ua, where each Ua is open in Aa and Ua = Aa for all but a finite number of indices a, is the usual basis taken for the Tychonoff topology [l; p. 98]. Perhaps a more natural way of defining a topology for X Aa would be to merely extend the definition of the box topology for finite products. The Goofynoff topology for X Aa is defined to be the topology generated by taking as a basis the set D of all subsets of X Aa of the form X Ua, where each Ua is a basis element of Aa. It is important to note that the sets generated in this manner are independent of the selection of bases for the factor spaces. We need to prove at this point that the set Dis indeed a basis for some topology on X Aa. Clearly the set D covers the space X Aa. Suppose B1, B2 E: D and g E: (Bl n B2). Then Bl = X ua and B2 = X Va for some collection of a Ua's and Va's where Ua and Va are basis elements in Aa for each E: A. E: t a) E: (U V ) for each a E: A. Thus Since g (B1 (\B2), we know tha g( a n a for each a E: A, there exists a basis elemen t Ya of Aa such that 4 g(a) E Ya C (Ua n Va). Hence X Ya is an element of Dsuch that g E X ya C (Bl n B2). Therefore, the set Dis a basis for some topology on X Aa [2; Th. 1.5, p. 22]. Given a product space S = X {Aa: aEA}, the Tychonoff topology on S will be denoted by T or just T (when no confusion arises) while S the Goofynoff topology will be denoted by G or just G. We will assume S that bases for the Aa are fixed and refer to the elements of the basis D for G as basic open sets. S Since XUa, where each Ua is open in Aa and Ua = Aa for all but a finite number of indices a, is clearly the union of basic open sets, we see that the Tychonoff topology is a subset of the Goofynoff topology for XAa. Consequently, any continuous function on (XAa,T) will also be continuous when viewed as a function on (XAa,G). Thus, for instance, the projection mapping TI : (XAa,G)�A is continuous S S th for each SEA. Furthermore, rr (XUa) = u and thus the s projection s s of a basic open set is a basis element of A implying that the S projection mappings on (XAa,G) are also open functions. There are several concepts in topology whose definitions vary slightly from text to text. We now give the version of each of these concepts which we will be using in this paper. A To-space is a space X such that, if p,q E Xand p # q, there exists an open set U with either p EU and qi U or pi U and q EU. A T1-space is a space X such that, if p,q EXand p # q, there exists an open set U with p EU and qi U.