Geodesic
Wallman extension and hyperspace. Functorial properties
Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 37-43 
H. O. Kukrak; V. L. Timokhovich. Wallman extension and hyperspace. Functorial properties. Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 37-43. http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Engelking R., Obschaya topologiya, Mir, M., 1986

[2] Harris D., “The Wallman compactification as a functor”, General Topology and its Applications, 1:3 (1971), 273–281 | DOI | MR | Zbl

[3] Hajek D., “A Characterization of $T_3$-spaces”, Indiana University Math. Journal, 23:1 (1973), 23–25 | DOI | MR | Zbl

[4] Kukrak G.O., “Ob otobrazheniyakh eksponentsialnykh prostranstv”, Vestnik BGU. Ser. 1, 1999, no. 3, 56–59 | MR | Zbl

[5] Loncar I., “Hyperspaces of the Inverse Limit Space”, Glasnik Matematicki, 27:47 (1992), 71–84 | MR | Zbl

[6] Fedorchuk V.V, Filippov V.V., Obschaya topologiya. Osnovnye konstruktsii, Fizmatlit, M., 2006

[7] Fedorchuk V.V., “O nekotorykh geometricheskikh svoistvakh kovariantnykh funktorov”, Uspekhi matematicheskikh nauk, 39:5 (1984), 169–208 | MR | Zbl

[8] Soban M., “Note sur la topology exponentialle”, Fund. Math., 71:5 (1971), 21–41 | MR