Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 1299-1307

Online first article October 16, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

Space of homeomorphisms under regular topology

Mir Aaliya, Sanjay Mishra

Lovely Professional University; Lovely Professional University


In this paper, we attempt to study several topological properties for the function space ${H(X)}$, space of self-homeomorphisms on a metric space endowed with the regular topology. We investigate its metrizability and countability and prove their coincidence at $X$ compact. Furthermore, we prove that the space ${H(X)}$ endowed with the regular topology is a topological group when $X$ is a metric, almost $P$-space. Moreover, we prove that the homeomorphism spaces of increasing and decreasing functions on $\mathbb R$ under regular topology are open subspaces of $H(\mathbb R)$ and are homeomorphic.

Keywords: Function space, regular topology, fine topology, metric space, homeomorphisms

MSC numbers: Primary 54C35, 54D99, 54D30, 54E35, 54H11

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