Compactness in the fine and related topologies

L. Holà

I will present results from the joint paper with professor R.A. McCoy "Compactness in the ne and related topologies".

Let X be a Tychonov space, Y a metrizable space and C(X,Y) be the space of contouous functions from X to Y. For a paracompact, locally hemicompact k-space X we characterize compact subsets of C(X,Y) topologized with the fine, graph and Krikorian topologies. Our results concerning compactness in the fine topology greatly generalized those of Spring [Topology Appl. 18 (1984) 87].

The following theorem is the main result of our paper.

Theorem. Let X be a paracompact, locally hemicompact k-space, (Y,d) a metric space and t be one of the following topologies: fine, graph, and Krikorian. The following are equivalent for a subset Q of C(X,Y):

(1 ) Q is countably compact in (C(X,Y),t);

(2) Q is compact in (C(X,Y),t);

(3) Q is sequentially compact in (C(X,Y),t);

(4) Q is almost compactly supported and Q is compact in (C(X,Y),t_co ), wthere t_co is the compact-open topology.

A subset Q of C(X,Y) is compactly supported provided that there exists a comnpact subset K of X such that for call f,g ÎQ, f | (X \ K)=g | (X \ K). A subset Q of C(X,Y) is almost compactly supported provided that it is the union of finitely many compactly supported subsets of C(X,Y).