Geodesic
On~$\varkappa$-metrizable space
Izvestiya. Mathematics , Tome 14 (1980) no. 2, pp. 407-440.

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In this paper the author studies spaces in which one can define a “distance” from points to canonically closed sets (the $\varkappa$-metric). It is proved that products of metric spaces and locally compact groups are examples of such spaces, and in these cases the $\varkappa$-metric can be constructed so that an analogue of the triangle axiom is satisfied. The topological structure of $\varkappa$-metrizable compact Hausdorff spaces is studied. Bibliography: 26 titles. 
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E. V. Shchepin. On~$\varkappa$-metrizable space. Izvestiya. Mathematics , Tome 14 (1980) no. 2, pp. 407-440. http://geodesic.mathdoc.fr/item/IM2_1980_14_2_a11/

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