International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 70671, 10 pages
doi:10.1155/2007/70671
Research Article
Mutually Compactificable Topological Spaces
Department of Mathematics, Faculty of Electrical Engineering and Communication, University of Technology, Technická 8, Brno 616 69, Czech Republic
Received 13 June 2006; Accepted 12 November 2006
Academic Editor: Lokenath Debnath
Copyright © 2007 Martin Maria Kovár. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Two disjoint topological spaces X, Y are (T2-) mutually compactificable if there exists a compact (T2-) topology on K=X∪Y which coincides on X, Y with their original topologies such that the points x∈X, y∈Y have open disjoint neighborhoods in K. This paper, the first one from a series, contains some initial investigations of the notion. Some key properties are the following: a topological space is mutually compactificable with some space if and only if it is θ-regular. A regular space on which every real-valued continuous function
is constant is mutually compactificable with no
S2-space. On the other hand, there exists a regular non-T3.5 space which is mutually compactificable with the infinite countable
discrete space.