Geodesic
A modification of the Harris ($WO$) condition and functor properties of the hyperspace and Wallman compactification
Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 4-11.

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A necessary and sufficient condition for the existence of a continuous extension $\text{exp} X \overset{\bar{f}}{\longrightarrow} \text{exp} Y$ of a map $X \overset{f}{\longrightarrow} Y$ is found, where $\text{exp} X$ is a hyperspace of the space $X$ endowed with Vietoris topology, and the map $\bar{f}$ is defined as $\bar{f}(F) = [f(F)]_Y$ ($[ \cdot ]_Y $ is a closure operator on the space $Y$). The obtained condition (named as $(\omega o)$ condition) is a modification of the Harris $(WO)$ condition. It is also shown, that ($\omega o$) condition is sufficient and in the case of regularity of a space $Y$ it is necessary for the existence of multivalued upper semi-continuous extension $\omega X \overset{\tilde{f}}{\longrightarrow} \omega Y$ of a map $f$ which satisfies some additional conditions ($\omega X$ is the Wallman compactification). The results obtained are commented on by category theory. 
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A. S. Biadrytski; V. L. Timokhovich. A modification of the Harris ($WO$) condition and functor properties of the hyperspace and Wallman compactification. Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 4-11. http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a0/

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