The closed graph and P-closed graph properties in general topology

On maps: continuous, closed, perfect, and with closed graph

“…A generalization and an analogue of theorem 5 of Piotrowski and Szymanski [3] and analogues of theorem 1.1.17 and corollary 1.1.18 of Hamlett and Herrington [4] are also obtained.…”

“…The definitions of subcontinuous and inversely subcontinuous maps can be found in Fuller MAIN RESULTS. THEOREM [4] . PROOF.…”

“…Theorem 6 includes theorem 16.19 of Thron ], while theorem 7 includes and gives analogues of corollary 1.1.18 of Hamlett and Herrington [4]. EXAMPLE.…”

Local characterization of functions with closed graphs

“…In [6] it is shown (for compact HausdorfF X and Hausdorff Y also in [13]) that a function / : X Y has a closed graph if and only if C(f, x) -{/(x)}, where C(f, x) is the cluster set of / at x defined by C(f, x) -Hi/gw CI /(?/) (= {y g Y : there.exists a net x a in X with limx a = x and lim/(z a ) = y}). Hence the following definition seems to be reasonable.…”

“…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.…”