Geodesic
On compact $T_1$-spaces
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2013), pp. 20-27

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We consider spaces, any subspaces of which are compact. We call such spaces hereditarily compact. The present work covers questions on the existence and methods of constructing hereditarily compact $T_1$-topologies. We prove the existence of $2^\tau$ pairwise incomparable hereditarily compact $T_1$-topologies on an infinite set $X$ of power $\tau$. The characteristics of hereditarily compact spaces are obtained. It is proved that the Tychonoff product of a finite number of hereditarily compact $T_1$-spaces is a hereditarily compact $T_1$-space, but the Tychonoff product of an infinite number of nonsingleton hereditarily compact $T_1$-spaces is not hereditarily compact.
Keywords: compactness, minimal $T_1$-topology, Tychonoff product. 
M. E. Voronov. On compact $T_1$-spaces. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2013), pp. 20-27. http://geodesic.mathdoc.fr/item/VUU_2013_3_a1/
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