On Mutual Compatificability of Topological Spaces
by
Martin Maria Kovár
Technical University of Brno

Recall topological space X is said to be \theta-regular [Ja] if every filter base in X with a \theta-cluster point has a cluster point. In Hausdorff spaces, \theta-regularity coincides with regularity. Further properties of \theta-regular spaces are also studied in [Ko]. Through this work, \theta-regularity plays a fundamental role. A topological space is said to be (strongly) locally compact if every x in X has a compact (closed) neighborhood. Compactness is regarded without any separation axiom. The following concepts will be introduced:

Definition Let X, Y be topological spaces with X \cap Y=\varnothing. The space X is said to be compactificable by the space Y or, in other words, X, Y are called mutually compactificable if there exists a compact topology on K = X \cup Y extending the topologies of X and Y such that any two points x in X, y in Y have disjoint neighborhoods in K. If, in addition, there exists a Hausdorff topology on K extending the topologies of X, Y we say that X is T₂-compactificable by Y or that X, Y are mutually T₂-compactificable.

Preliminary observations

1. A real interval is T₂-compactificable by any real interval.
2. A discrete space is T₂-compactificable by a copy of itself.
3. A space is compactificable by a finite discrete space iff the space is strongly locally compact.
4. For a space X there exist a space Y which is T₂-compactificable by X iff X is T_3¹/2.

We intend to discuss some variants the of concepts defined above and also some of the following natural questions:

1. Characterize all topological spaces X such that there exists a space Y such that X, Y are mutually compactificable.
2. Characterize those topological spaces X that are (T₂-) compactificable by some fixed space Y.
3. Characterize those topological spaces that are (T₂-) compactificable by a copy of itself.

Ja Jankovi\'c D. S. \theta-regular spaces preprint published: Internat. J. Math. Sci. 8 1985 no.3, 615-619

Ko Kovár M. M. On \theta-regular spaces Internat. J. Math. Sci. 17 1994 687-692

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-57.