On compact $T_1$-spaces

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.