Discontinuous actions on cones, joins, and $n$-universal bundles

Alexandru Chirvasitu

Paper

Discontinuous actions on cones, joins, and $n$-universal bundles

Abstract

We prove that locally countably-compact Hausdorff topological groups

act continuously on their iterated joins

(the total spaces of the Milnor-model

-universal

-bundles), and the converse holds under the assumption that

is first-countable. In the latter case other mutually equivalent conditions provide characterizations of local countable compactness: the fact that

acts continuously on its first self-join

, or on its cone

, or the coincidence of the product and quotient topologies on

for all spaces

or, equivalently, for the discrete countably-infinite

. These can all be regarded as weakened versions of

's exponentiability, all to the effect that

preserves certain colimit shapes in the category of topological spaces; the results thus extend the equivalence (under the separation assumption) between local compactness and exponentiability.