Properties of locally H-closed spaces

Abstract:

This paper investigates the properties of locally $H$-closed spaces with regard to extensions, subspaces, and functions. We solve the $H$-closed extension remainder problem by showing that a space is locally $H$-closed if and only if it has a $\theta$-closed remainder in some $H$-closed extension. In fact, an $H$-closed space is Urysohn iff every $H$-closed subspace is $\theta$-closed. We solve the locally $H$-closed subspace problem by giving a necessary and sufficient condition for a subspace of a locally $H$-closed space to be locally $H$-closed. In particular, an open subspace of a locally $H$-closed space is locally $H$-closed if and only if its boundary is a $\theta$-closed subspace of its closure. An $H$-closed space is shown to be compact if and only if every open subset is locally $H$-closed. A retract of a locally $H$-closed space is locally $H$-closed.