Wallman extension and hyperspace. Functorial properties

D. Harris introduced the concept of a $WO$-map and proved that any $WO$-map $X\xrightarrow{f}Y$ admits a continuous extension $\omega X\xrightarrow{\widetilde{f}}\omega Y$ ( $\omega X$– Wallman compactification of the space $X$). The paper investigates modifications of the condition $(WO)$ ($WO(2)$, $WO(2$-$2)$, $WO(comb)$). It is shown that any $WO(2$-$2)$-mapping $X\xrightarrow{f}Y$ ($X$ and $Y$ are $T_1$-spaces) admits a continuous extension to the mapping $\exp X\xrightarrow{\overline{f}}\exp Y$ ($\exp X$ is a hyperspace of the space $X$ with a Vietoris topology), and if $X$ and $Y$ are regular and $f$ is a $WO$-mapping, then $f$ can be continuous extended to the mapping ${{\exp }^{n}}\omega X\xrightarrow{{{f}_{n}}}{{\exp }^{n}}\omega Y$ (${{\exp }^{n}}\omega X=\underbrace{\exp ...\exp }_{n}\omega X,\ n\in \mathbb{N}$ ). Thus, on the categories $\mathcal{K}_1$ of $T_1$-spaces and $WO(2$-$2)$- maps and $\mathcal{K}_2$ of $T_3$-spaces and $WO$-maps, the covariant functors $\exp $ and ${{\exp }^{n}}\omega$ are defined respectively.