International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 16135, 11 pages
Research Article

The Classes of Mutual Compactificability

Martin Maria Kovár

Department of Mathematics, Faculty of Electrical Engineering and Communication, University of Technology, Technická 8, Brno 616 69, Czech Republic

Received 17 June 2004; Revised 28 November 2005; Accepted 13 February 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.


Two disjoint topological spaces X, Y are mutually compactificable if there exists a compact topology on K=XY which coincides on X, Y with their original topologies such that the points xX, yY have disjoint neighborhoods in K. The main problem under consideration is the following: which spaces X, Y are so compatible such that they together can form the compact space K? In this paper we define and study the classes of spaces with the similar behavior with respect to the mutual compactificability. Two spaces X1, X2 belong to the same class if they can substitute each other in the above construction with any space Y. In this way we transform the main problem to the study of relations between the compactificability classes. Some conspicuous classes of topological spaces are discovered as the classes of mutual compactificability. The studied classes form a certain “scale of noncompactness” for topological spaces. Every class of mutual compactificability contains a T1 representative, but there are classes with no Hausdorff representatives.