Document Type : Original Manuscript

**Authors**

Department of Mathematics, Yazd University, P.O. Box 89195 - 741, Yazd, Iran.

**Abstract**

Abstract. A topological group H is called ω -narrow if for every

neighbourhood V of it’s identity element there exists a countable

set A such that V A = H = AV. A semigroup G is called a generalized group if for any x ∈ G there exists a unique element e(x) ∈ G

such that xe(x) = e(x)x = x and for every x ∈ G there exists

x − 1 ∈ G such that x − 1x = xx − 1 = e(x). Also let G be a topological space and the operation and inversion mapping are continuous,

then G is called a topological generalized group. If {e(x) | x ∈ G} is

countable and for any a ∈ G, {x ∈ G|e(x) = e(a)} is an ω-narrow

topological group, then G is called an ω-narrow topological generalized group. In this paper, ω-narrow and resolvable topological

generalized groups are introduced and studied

**Keywords**

groups, Journal of Mahani Mathematical Research Center, 5 (2016), 27–32.

[2] M. R. Ahmadi Zand and S. Rostami, Some topological aspects of generalized groups and pseudonorms on them, Honam Math. J., 40 (2018), 661–669.

[3] A. Arhangelskii and M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, Atlantis Press/World Scientific, 2008.

[4] W. W. Comfort and J. van Mill, Groups with only resolvable group topologies, Proc. Amer. Math. Soc., 120 (1994), 687–696.

[5] R. Engelking, General topology, Revised and Completed Edition, Heldermann

Verlag, Berlin, 1989.

[6] E. Hewitt, A problem of set-theoretic topology, Duke Math. J., 10 (1943),

309–333.

[7] M. R. Mehrabi, M. R. Molaei and A. Oloomi, Generalized subgroups and homomorphisms, Arab. J. Math. Sci., 6 (2000), 1–7.

[8] M. R. Molaei, Generalized groups, Bull. Inst. Pol. Din. Iasi, 65 (1999), 21–24.

[9] M. R. Molaei, Topological generalized groups, Int. J. Pure Appl. Math., 9 (2000), 1055–1060.

[10] M. R. Molaei, Mathematical structures based on completely simple semigroups, Hadronic Press Monographs in Mathematics, Hadronic Press, USA, 2005.

[11] J. R. Munkres, Topology, Second edition, Prentice Hall, 2000.

[12] G. R. Rezaei and J. Jamalzadeh, The continuity of inversion in topological generalized group, General Mathematics, 20 (2012), 69–73.