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On the functor properties of some hyperspace topologies
A. S. Bedritskiy,
V. L. Timokhovich Belarusian State University, 4 Nezavisimosti Ave., Minsk, 220030 Belarus
Abstract:
A continuous map
$X \overset{f}{\longrightarrow} Y$ and its extension $\exp_{\tau} X \overset{\bar{f}}{\longrightarrow} \exp_{\tau}Y$ are considered (
$\exp_{\tau} X$ is the hyperspace (endowed with a topology
$\tau$) of the topological space
$X$,
$\bar{f}(F) = [f(F)]_Y$ (the closure of a set
$f(F)$ in the space
$Y$)). A necessary and sufficient condition (a modification of the Harris' (WO) condition) of continuity of the map
$\bar{f}$ in the cases when
$\tau = \tau_{LF}$ (the locally finite topology) and
$\tau = \tau_F$ (the Fell topology) is found. When
$X$ and
$Y$ are metrizable spaces the topology
$\tau_{\inf}$, as the infimum of all Hausdorff metric topologies, is considered. A sufficient condition (
$(TUC)$ condition) of continuity of the map
$\bar{f}$ in the case when
$\tau =\tau_{\inf}$ is found. It is also shown, that this condition is necessary, when the space
$Y$ is locally compact and second countable. The results are commented from the point of view of the category theory.
Keywords:
hyperspace, Fell topology, locally finite topology, Hausdorff metric topology, infimum topology.
UDC:
515.12 Received: 29.01.2024
Revised: 29.01.2024
Accepted: 26.06.2024
DOI:
10.26907/0021-3446-2025-2-15-28
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