The open-point and bi-point-open topologies on C(X)

Abstract

In the definition of a set-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, we use a certain family of subsets of X and open subsets of R. But instead of using this traditional way to define topologies on C(X), in this paper, we adopt a different approach to define two interesting topologies on C(X). We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies.

The bi-compact-open topology on C(X)

“…Here U is some given base for X and V is some given countable base for R consisting of bounded open intervals. Now Proposition 2.1 in [6] implies the following proposition.…”

“…Consider the neighborhood ( Remark 1. In [6], it is shown that the space C ph (X) is completely regular if and only if X 0 is G δ -dense in X if and only if C h (X) is completely regular. Since X 0 ⊆ lc(X), complete regularity of the space C ph (X) implies the complete regularity of the space C kh (X).…”

“…In [6] and [7], we have shown that the space, C ph (X) has some nice properties. For example, the pseudocharacter of C ph (X) is equal to the density of X; that is, ψ(C ph (X)) = d(X).…”

Completeness properties of the open-point and bi-point-open topologies on C(X)

“…(a) ⇒ (j). If C h (X) is completely metrizable, then Theorem 3.7 in [9] implies that X 0 is G δ -dense in X. Hence by Theorem 2.3, X is a countable discrete space.…”

“…By Theorems 3.7 and 3.8 in [9], if X 0 is G δ -dense in X, then the spaces C h (X) and C ph (X) are topological groups and hence homogeneous. A space X is called homogeneous if for every pair of points x, y in X, there exists a homeomorphism of X onto X itself which carries x to y.…”