c-continuous fundamental groups

A unified approach to continuous and certain non-continuous functions II

“…The results obtained in the process improve and unify scores of known results on various non-continuous functions, and often suggest new results. [12] pertaining to c-continuous functions and with P = Lindelofness it yields Theorem 2.9 of [23] corresponding to ^-continuous functions. Similarly, for P = countable compactness it gives a sufficient condition for a c*-continuous function to be continuous.…”

“…Similarly, for P = countable compactness it gives a sufficient condition for a c*-continuous function to be continuous. Hoyle [12] concerning c-continuous functions and for P = the Lindelof property it gives Theorem 2.10 of [23] pertaining to ^-continuous functions.…”

“…In [19] we developed a unified and coherent theory of continuous and certain noncontinuous functions and introduced the notions of P-continuous functions and semi-P-functions (or P-proper maps) as unifying tools. It turns out that the theory of P-continuous functions encompasses in one the theories of continuous functions, upper (lower) semicontinuous functions, c-continuous functions [12], almost continuous functions [51], c*-continuous functions [46], s-continuous functions [20], .ff-continuous functions [30], ^-continuous functions [23], Ti-continuous functions [1], £-continuous functions ( [17,18]), and several other generalisations of continuity. In the same vein the theory of P-proper maps (-semi P-functions) provides a unified framework for dealing with continuous functions, ^-continuous functions [44], semiconnected functions ( [16,28]), ^-continuous functions [53], strongly c-continuous functions [10], stronglŷ -continuous functions [11], R-m&ps ( [4,45]), and other similar variants of continuity.…”

Local characterization of functions with closed graphs

“…A function is c-continuous if it is such at every point [5]. Denote by T(f) the set of all c-continuity points of /.…”

“…It is known that if Y is compact [8], [6] or if X is first countable and Y is countably compact [9], [6] or if X is saturated and Y is regular countably compact [5], then functions with closed graphs are continuous. However, from their proofs it follows that under above assumptions on X and We recall that a topological space is almost resolvable if it is a countable union of sets with empty interiors.…”