How to Generalize the Notion of a k-Space
by
R. Bartsch
Universität Rostock
Coauthors: H. Poppe

Let (X, \tau) be a topological space and E a property, which is defined for subsets of (X, \tau). Any dependence of E on the underlying topology is not required. We denote the family of all subsets of X, which have property E under \tau as E(X, \tau). We call a topological space (X, \tau) to be E-generated iff every subset A of X which has an open intersection with all elements of E(X, \tau) in their trace topolgies, is open in (X, \tau). If E means compactness, we find in this way the usual k-spaces (with or without Hausdorff axiom). For these we know

Theorem. (D. E. Cohen) A Hausdorff space X is a k-space iff X is a quotient space of a locally compact space.

We find a similar result for a class of properties E.

Furthermore we consider the problem of the E-extension \tauE of a given topology \tau, i. e. the family of all subsets A of X with open intersection in all elements of E(X, \tau). We are especially interested to decide whether or not E(X, \tau) = E(X, \tauE) holds.

1 Dugundji, J. Topology Allyn and Bacon, Inc., Boston, 1966
2 Poppe, H. On locally defined topological notions Q & A in General Topology 13 1995

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 # caaj-26.