Topology Atlas
Document # ppae-21

## Proper and admissible topologies in the setting of closure spaces

### Mila Mrsevic

Proceedings of the Ninth Prague Topological Symposium
(2001)
pp. 205-216
A Cech closure space (X, u) is a set X with a
(Cech) closure operator u which need not be idempotent.
Many properties which hold in topological spaces hold in Cech closure
spaces as well.

The notions of proper (splitting) and admissible (jointly continuous)
topologies are introduced on the sets of continuous functions between
Cech closure spaces. It is shown that some well-known results of Arens
and Dugundji and of Iliadis and Papadopoulos are true in this setting.

We emphasize that Theorems 1-10 encompass the results of A. di Concilio
and of Georgiou and Papadopoulos for the spaces of continuous-like
functions as \theta-continuous, strongly and weakly
\theta-continuous, weakly and super-continuous.

Mathematics Subject Classification. 54A05 54A10 54C05 54C10.

Keywords. Cech closure space, function space, proper
(splitting) topology, admissible (jointly continuous) topology,
$\theta$-closure, $\theta$-continuous function, strongly
$\theta$-continuous, weakly $\theta$-continuous, weakly continuous,
super-continuous function.

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Copyright © 2002
Charles University and
Topology Atlas.
Published April 2002.