Cauchy Spaces. I. Structure and Uniformization Theorems.
We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness...
We prove that if is a union of subspaces of pointwise countable type then the space is of pointwise countable type. If is a countable union of ultracomplete spaces, the space is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].
The problem whether every topological space has a compactification such that every continuous mapping from into a compact space has a continuous extension from into is answered in the negative. For some spaces such compactifications exist.
Suppose that P is a finite 2-polyhedron. We prove that there exists a PL surjective map f:Q → P from a fake surface Q with preimages of f either points or arcs or 2-disks. This yields a reduction of the Whitehead asphericity conjecture (which asserts that every subpolyhedron of an aspherical 2-polyhedron is also aspherical) to the case of fake surfaces. Moreover, if the set of points of P having a neighbourhood homeomorphic to the 2-disk is a disjoint union of open 2-disks, and every point of P...
We study relations between the cellularity and index of narrowness in topological groups and their -modifications. We show, in particular, that the inequalities and hold for every topological group and every cardinal , where denotes the underlying group endowed with the -modification of the original topology of and is the index of narrowness of the group . Also, we find some bounds for the complexity of continuous real-valued functions on an arbitrary -narrow group understood...
Given a discrete group , we consider the set of all subgroups of endowed with topology of pointwise convergence arising from the standard embedding of into the Cantor cube . We show that the cellularity for every abelian group , and, for every infinite cardinal , we construct a group with .