Vol. 32, No. 2, 1970

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On continuity conditions for functions

Evelyn Rupard McMillan

Vol. 32 (1970), No. 2, 479–494

Suppose f is a function from a topological space X into a topological space Y. Many classical theorems of topology assert that from continuity of f follow certain other properties. For example, if f is continuous, then its point inverses must be closed (if Y is a T1 space) and compact subsets of X must have compact images under f, that is, f must be compact preserving. Also, f must be connected, that is, connected subsets of X have connected images. It is natural to ask whether some combination of these properties is equivalent to continuity. In particular, our main result (Theorem 2) concerns compact preserving, connected functions. We give a proof of a result announced but inadequately proved by E. Halfar that such a function f is continuous if X is a locally connected Hausdorff space with property K and Y is a Hausdorff space. (A space X has property K if given that x0 is a limit point of an infinite set A X, then there is a compact set K A ∪{x0} such that x0 is a limit point of K.)

Mathematical Subject Classification
Primary: 54.60
Received: 22 January 1969
Published: 1 February 1970
Evelyn Rupard McMillan