Homology properties of arbitrary subsets of Euclidean spaces
Samuel Kaplan · Transactions of the American Mathematical Society · 1947
Introduction.The study of the homology properties of topological spaces by means of coverings, first clearly formulated by Cech [3](1), has been successfully applied to compact spaces by(2) him and other topologists.The large body of theory which has grown up around this work, however, deals almost exclusively with finite coverings, and these alone are not a sufficient tool for non-compact spaces.This was graphically demonstrated by Dowker [4], who showed that under finite coverings the linear continuum has nonbounding one-dimensional Cech cycles.Essentially the trouble is that we cannot expect to make a thorough analysis of a topological space, as far as homology is concerned, without using a complete family of coverings; and for a non-compact space the finite coverings do not form a complete family.Thus, if we are going to extend our studies to spaces other than the compact, a basic problem in the study of each type of space will be to find the simplest possible complete family of coverings for that type.In the present paper we take up the class of separable metric spaces.Since these include all the subsets of Euclidean spaces, they constitute a large and important collection.The finite coverings of course do not form a complete family for such spaces.Next to the finite, the simplest coverings are the countable star-finite, and we show in §1 that these do form a complete family.In §2 we consider Cech cycles on compact subsets of a separable metric space, and compare the property of bounding on a compact subset with the (more general) property of bounding in the space as a whole.We obtain a topologically invariant type of bounding for Vietoris cycles which is equivalent to the latter.In §3 we consider the question of obtaining the homology properties of an arbitrary subset A of a separable metric space R by means of open sets of R. As is well known, if A is closed in R, we can obtain all the homology properties of A by using coverings of R and considering the parts of such coverings which meet A. If A is not closed, this is no longer true.As is shown in §3, the general solution is to use not coverings of R but coverings of all the neighborhoods of A in R.In §4 we establish the isomorphism between the Cech homology groups Presented to the Society, April 18, 1942, under the title Homologies in metric separable spaces; received by the editors August 30, 1946.(l) Numbers in brackets refer to the references cited at the end of the paper.(') A space R will be called compact if for every covering of R by open sets, there is a finite number of the open sets which covers If.