OPEN-POINT AND BI-POINT OPEN TOPOLOGIES ON CONTINUOUS FUNCTIONS BETWEEN TOPOLOGICAL (SPACES) GROUPS

Authors

  • B. K. Tyagi Department of Mathematics, Atmaram Sanatan Dharma College, University of Delhi, New Delhi-110021, India Author
  • S. Luthra Department of Mathematics, University of Delhi, New Delhi-110007, India Author

Keywords:

Point-open topology, open-point topology, bi-point-open topology, topological group, zero dimensional, ω-narrow, disjoint shrinking, discrete selection

Subjects:

54C35, 54A10, 54C05, 54D10, 54D15, 54E35, 54H11

Abstract

In this paper, we study the notions of point-open topology Cp(X,H), open-point topology Ch(X,H) [resp. Ch(G,H)] and bi-point-open topology Cph(X,H) [resp. Cph(G,H)] on C(X,H) [resp.C(G,H)], the set of all continuous functions from a topological space X (topological group G) to a topological group H. In this setting, we study the countability, separation axioms and metrizability. The equivalent conditions are given so that the space Ch(G,H) is a zero-dimensional topological group. Further, if G is H-regular, then Ch(G,H) is Hausdorff if and only if G is discrete. It is shown that under certain conditions the topological groups Cp(X,H), Ch(X,H) and Cph(X,H) are ω-narrow. Sufficient conditions are given for the topological spaces Cp(X,H), Ch(X,H) and Cph(X,H) to be discretely selective and to have a disjoint shrinking.

Published

2022-01-15