ADV MATH SCI JOURNAL Advances in Mathematics: Scientific Journal 9 (2020), no.7, 4787–4795 ISSN: 1857-8365 (printed); 1857-8438 (electronic) https://doi.org/10.37418/amsj.9.7.45 * G ˆ α-HOMEOMORPHISM IN TOPOLOGICAL SPACES K. M. ARIFMOHAMMED 1 , S. PIOUS MISSIER, AND M. SURAIYA BEGUM ABSTRACT. In this article, we introduce a special type of generalised closed and open maps, briefly * g ˆ α-closed maps, * g ˆ α-open maps in topological spaces also analyze some important characterizations of these new type of maps and then we study * g ˆ α-homeomorphisms. We also obtain strongly * g ˆ α-homeomorphisms and proved that under the operation ◦ the collection of all strongly * g ˆ α- home- omorphisms form a group. 1. I NTRODUCTION In topological space, R. Malghan [10] introduced and investigated the no- tion of generalised-closed maps. In 1994, R. Devi [5] introduced the notions of semi-generalized-closed maps and generalized semi-closed maps. In topolog- ical space, many authors have been introduced different types of generalized homeomorphisms. In 1972, Crossely and Hildebrand [2] have analyzed semi- homeomorphisms. In 1991, Maki et al [9] have characterized the notion of generalized-homeomorphisms. In our present study,we introduce a special type of generalised closed and open maps, briefly * g ˆ α-closed maps, * g ˆ α-open maps in topological spaces also analyze some important characterizations of these new type of maps and then we study * g ˆ α-homeomorphisms. We also obtain strongly 1 corresponding author 2010 Mathematics Subject Classification. 54A05, 54D01, 54F65, 54G05. Key words and phrases. * g ˆ α-closed maps, ∗g ˆ α - open map, * g ˆ α-homeomorphisms, strongly * g ˆ α-homeomorphisms. 4787