In this paper a number of
conditions on a function from one topological space to another are considered.
Among these conditions are those of a function or its inverse preserving
closedness, openness, or compactness of sets. Other conditions are having a
closed graph and a concept generalizing continuity, subcontinuity, which we
introduce.
Some interesting results which are uncovered are the following: (1) A function
which is closed with closed point inverses and a regular space for its domain has a
closed graph. (2) If a function maps into a Hausdorff space, continuity of the function
is equivalent to the requirement that the function be subcontinuous and
have a closed graph. (3) The usual net characterization of continuity for a
function with values in a Hausdorff space is still valid if it is required only that
the image of a convergent net be convergent (not necessarily to the “right”
value).
