ON MAPS : CONTINUOUS , CLOSED , PERFECT , AND WITH CLOSED GRAPH

This paper gives relationships between continuous maps, closed maps, perfect maps, and maps with closed graph in certain classes of topological spaces.


INTRODUCTION.
Throughout, by a space we shall mean a topological space.No separation axioms are assumed and no map is assumed to be continuous or onto unless mentioned explicitly; cl(A) will denote the closure of the subset A in the space X.A space X is said to be Tlat its subset _A if each point of A is closed in X. X is said to be B-W Compact 1I.!1 if every infinite subset of X has at least one limit point.A point x in X is said to be a cluster point limi___t in the terminology of Thron 1]) of a subset A of X if every neighbourhood of x contains an infinite number of points of A. X is said to be a Frechet space if whenever x e cl(A), there is a sequence of points in A converging to x.A map f:X Y is said to be perfect if it is continuous, closed, and has compact fibers f-1 (y), y ey.For study of perfect maps, see [2] and its references.
The prima_,'y purpose of this paper is to give relationships between continuous maps,closed maps, perfect maps, and maps with closed graph.A generalization and an analogue of theorem 5 of Piotrowski and Szymanski [3] and analogues of theorem 1.1.17and corollary 1.1.18 of Hamlett and Herrington [4]   are also obtained.
NOTE.The definitions of subcontinuous and inversely subcontinuous maps can be found in Fuller MAIN RESULTS.THEOREM [4] .Let f:X Y be continuous, where Y is Hausdorff.Then f has closed graph.THEOREM 2. Let f:X Y be closed with closed (compact) fibers,where X is regular (Hausdorff).Then f has closed graph.
PROOF.We prove only the parenthesis part; the other part, which can also be proved in a simple manner by using our proof of the parenthesis part, has been proved by Fuller [5,corollary 3.9]and by Hamlett and Herrington [4, theorem 1.1.17]by different techniques.Let xeX, yeY, yCf(x).Then xf-l(y), which is compact.Since X is Hausdorff, there exist disjoint open sets U and V containing x and f-1 (y) respectively.Then f is closed implies there exists an open set W containing y such that f-(W)cV and therefore, f(U)W=.It follows that f has closed graph.
Combining theorems and 2, we get the following THEOREM 3. Let f:X Y be perfect, where either X or Y is Hausdorff.Then f has closed graph.
The following theorem 4(theorem 5), part (b) of which is a generalization (analogue) of theorem 5 of Piotrowski and Szymanski [3], gives sufficient conditions under which the converse of theorem (theorem 2) holds.PROOF.We give the proof of part (b) only; part (a) is well known (corollary 2(b) of Piotrowski and Szymanski [3],and theorem 1.1.10 of [4]), while part (c) is theorem 3.4 of Fuller [5].Let F be a closed subset of Y and let xeclf-l(F)-f-I(F).Since X is a Frechet space, there exists a sequence {Xn} of points in f-(F) such that x n x.Since f has closed graph, the set H of values of the sequence f(xn) is an infinite subset of the B-W compact set F and F is T at H. Therefore, H has a cluster point yeF, y f(x), and the set U=X-f-1(y) is an open set containing x. Then x n x implies there exists a positive integer n o such that xneU for all n_>n o.Again fhas closed graph and the set K={xn:n_>no}U{x} is compact; it follows that f(K) is closed, which is a contradiction since it is easy to see that yeclf(K)-f(K).Hence f must be continuous.
THEOREM. 5. Let f:X Y have closed graph.Then f is closed if any one of the following conditions is satisfied.
(a) X is compact, (b) X is countably compact and Y is Frechet, (c) f is inversely subcontinuous.
PROOF.We give the proof of part (b) only; part (a) is well known (corollary 2(a) of Piotrowski and Szymanski [3]), while part (c) is theorem 3.5 of Fuller [5].Let F be a closed subset of X and let yeclf(F)-f(F).Since Y is Frechet and T at f(X), there exists a sequence f(xn) }of distinct points converging to y where x nF.Now the set of values of the sequence x n is an infinite subset of the countably compact set F and therefore, it has a cluster point xeF, y f(x).Since Y is T at f(X), the set V =Y-{ f(x) is an open set containing y. Then f(xn) y implies there exists a positive integer n o such that f(Xn)eV for all n_> o.Since fhas closed graph and the set K ={f(Xn):n_.>no}U{y is compact, it follows that f-I(K) is closed, which is a contradiction since it is easy to see that xeclf-l(K)-f-I(K).Hence f must be closed.
Combining theorems and 5(theorems 2 and 4), we obtain the following theorem 6 (theorem 7), giving a relationship between continuous and closed maps.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].THEOREM 6.Let f:X Y be continuous, where Y is Hausdorff and one of the conditions (a), (b), (c) in theorem 5 is satisfied.Then f is closed.The condition that X is countably compact in theorems 5(b)and 6(b) cannot be replaced by the weaker condition that X is B-W compact,as the following example shows.EXAMPLE.Let X=N, the positive integers, with a base for a topology on X the family of all sets of the form {2n-l,2n},neN, and Y={0,1,1/2 l/n as a subspace of the real line.The map f:X Y, defined by f(2n-1)=l/n-l=f(2n) for >n.2 and f(1)---0=f(2), is a continuous surjection which is not closed, although X is B-W compact and Y is Frechet, Hausdorff.

THEOREM 4 .
Let f:X Y have closed graph.Then f is continuous if any one of the following conditions is satisfied.(a) Y is compact, (b) X is Frechet and Y is B-W compact, (c) f is subcontinuous.