SUFFICIENT CONDITIONS FOR CONTINUITY OF MAPS

A function f : X → Y is said to have closed graph if graph of f , i.e. the set (x, f(x)) is a closed subset of the product space X × Y . It is well established fact that a function with closed graph is KC as well as inversely KC. Moreover, a function with closed graph is continuous if Y is compact. In the present paper, some conditions are investigated under which an inversely KC map or KC map becomes continuous. AMS Subject Classification: 54C05


Introduction
By a space, we shall mean a topological space.No separation axioms are assumed and no function is assumed to be continuous or onto unless mentioned explicitly; cl(A) will denote the closure of the subset A in the space X.If A is a subset of X, we say that X is T 1 at A if each point of A is closed in X.A point x in X is said to be a cluster point (w-limit point in the terminology of Thron [4])of a subset A of X if every neighborhood of x contains infinite number of points of A. X is said to be Frechet space (or closure sequential in Received: June 19, 2014 c 2014 Academic Publications, Ltd. url: www.acadpubl.euthe terminology of Wilansky [5]) if for each subset A of X, x ∈ cl(A) implies there exists a sequence {x n } in A converging to x. X is said to be a k-space if O is open (equivalently:closed) in X whenever O ∩ K is open (closed) in K for every compact subset K of X.Every space which is either locally compact or Frechet is a k-space.
A function f : X → Y is said to be compact preserving (compact) if image (inverse image) of each compact set is compact.f : X → Y will be called KC [5] (inversely KC) if image (inverse image) of every compact set is closed.
Theorem 1.2.Let f : X → Y have closed graph and be compact preserving where X is a k-space.Then f is continuous.
A function with closed graph is KC as well as inversely KC.In the present paper, the condition of closed graph on the map f in theorem 1.1 and theorem 1.2 is weakened by assuming the map KC or inversely KC and conditions are investigated under which a KC map (inversely KC map) becomes continuous.

Main Results
The following theorems 2.1 and 2.2 give conditions under which a KC map becomes continuous.
Theorem 2.1.Let f : X → Y be KC, compact preserving and has closed point inverses where X is a Frechet.Then f is continuous.
is an open set containing x. Then x n → x implies there exists an integer n 0 such that x n ∈ U for all n ≥ n 0 .Let K = {x n : n ≥ n 0 } ∪ {x}.Then K is compact subset of X , but f (K) is not closed as y ∈ clf (K) − f (K) -a contradiction.Hence f must be continuous.
In the next Theorem 2.2, the condition of compact preserving on the map f is replaced by taking Y as B-W compact.
Theorem 2.2.Let f : X → Y be KC and have closed point inverses where X is Frechet and Y is B-W compact.Then f is continuous.
Proof.For the proof of this theorem, see [2].
The following theorems 2.3 and 2.4 give conditions under which an inversely KC map becomes continuous.
Theorem 2.3.Let f : X → Y be inversely KC and compact preserving where X is a k-space.Then f is continuous.
is closed in K as f is inversely KC.This completes the proof.
In the next Theorem 2.4 the condition of compact preserving on the map f is dropped and Y is assumed as locally compact, regular, countable compact and T 1 at f (X).
Theorem 2.4.Let f : X → Y be inversely KC, where X is Frechet and Y is locally compact, regular, countable compact and T 1 at f (X).Then f is continuous.
Proof.Let F be a closed subset of Y and let ∈ cl(W ).Since f is inversely KC, f −1 (clW ) is closed which is a contradiction as x ∈ clf −1 (clW ) − f −1 (clW ).Hence f must be continuous.
The following example shows that none of the condition on the domain and range space in theorems 2.1 to 2.4 can be weakened.
x) is closed .Now Y is locally compact, regular implies there exists an open set W containing y such that cl(W ) is compact and f (x) /