A topological property of
subspaces of a Hausdorff space, called 𝜃-closed, is introduced and used to prove and
interrelate a number of different results. A compact subspace of a Hausdorff space is
𝜃-closed, and a 𝜃-closed subspace of a Hausdorff space is closed. A Hausdorff space X
with property that every continuous function from X into a Hausdorff space is closed
is shown to have the property that every 𝜃-continuous function from X into a
Hausdorff space is closed. Those Hausdorff spaces in which the Fomin H-closed
extension operator commutes with the projective cover (absolute) operator are
characterized. An H-closed space is shown not to be the countable union of 𝜃-closed
nowhere dense subspaces. Also, an equivalent form of Martin’s Axiom in
terms of the class of H-closed spaces with the countable chain condition is
given.
