On $H$-closed topological semigroups and semilattices

In this paper, we show that if $S$ is an $H$-closed topological semigroup and $e$ is an idempotent of $S$, then $eSe$ is an $H$-closed topological semigroup. We give sufficient conditions on a linearly ordered topological semilattice to be $H$-closed. Also we prove that any $H$-closed locally compact topological semilattice and any $H$-closed topological weakly $U$-semilattice contain minimal idempotents. An example of a countably compact topological semilattice whose topological space is $H$-closed is constructed.