Mosaics of compact metric spaces

Abstract

Introduction. Our purpose is to study a class of topological spaces which is more general than the class of first countable Hausdorff spaces and less general than the class of k-spaces. To construct a space (X, 3) of this class, called a mosaic space, there is employed a collection of topological spaces (Xa, 3a): aGA } which cover X and which serve as test to determine the topology 3. Since each space (Xa, 3a) iS to be thought of as the prototype of a fairly geometrical object such as an arc, curve, or simplex, we require that each (Xa, 3a) be a compact metric space; in order to guarantee that each such compact metric space be a subspace of (X, 3) the former are patched together in a suitable fashion. After establishing the definition of a mosaic space and some basic properties, we show that among the topological spaces it is precisely the mosaic spaces which are in one-to-one correspondence with the Kuratowski ?*-spaces which have the additional feature that limits are unique. By generalizing the mosaic concept by patching together functions we find that a closed continuous image of a mosaic space is a mosaic space; that a quasi-compact image of a mosaic space is again a mosaic space, provided limits of convergent sequences are unique in the image space; and that every mosaic space is the quasi-compact image of a suitable locally compact metric space. The latter half of the paper gives necessary and sufficient conditions for a mosaic space to be hereditary, that is, for each subspace to be a mosaic space; for a mosaic space to be countably compact; and for the Cartesian product of a fixed mosaic space with an arbitrary mosaic space to be again a mosaic space. Further it is shown that a mosaic space can always be countably compactified so as to remain a mosaic space, and necessary and sufficient conditions are given for a mosaic space to be embedded in a compact mosaic space. Also we give an example of a compact hereditary mosaic space which is not a Hausdorff space. Finally, some results are obtained for function space topologies within the mosaic context. 1. Definition and properties of mosaic spaces. There will frequently arise topologies 3 which are defined on subsets Y of a set X rather than on X itself. Moreover, closed sets will be of greater importance than open sets. We thereby adopt for the expression EE5 the following definition: for every