M. R. Ahmadi Zand - Department of Mathematics, Yazd University, P.O. Box ۸۹۱۹۵ - ۷۴۱, Yazd, Iran.
S. Rostami - Department of Mathematics, Yazd University, P.O. Box ۸۹۱۹۵ - ۷۴۱, Yazd, Iran.

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

ω-narrow topological generalized group, Resolvable topological generalizad group, Precompact topological generalized group, Invariance number