Vol. 25, No. 3, 1968

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Relations among continuous and various non-continuous functions

Richard Vernon Fuller

Vol. 25 (1968), No. 3, 495–509

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).

Mathematical Subject Classification
Primary: 54.60
Received: 23 August 1967
Published: 1 June 1968
Richard Vernon Fuller