We introduce some new concepts of topological spaces which say a-separable topological space and O-topological group, a-first axiom, a-second axiom, and we find some relations between them with some applications in normed spaces.

We show that the lattice of compactifications of a topological group G is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification bG of G. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained

Given a discrete semi-group $S$, we define a topological action $\theta$ of S on a locally compact space $X$. Additionally, there is an action α of $S$ on the $C^∗$-algebra $C0(X)$, which is introduced in relation to $θ$. We explore the topological independence of $\theta$ and the impact of $\theta$ on α. Finally, we discuss the concept of a semi-partial dynamical system $(C0(X), S, \alpha)$ and examine some of its properties.

The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops L of dimension  3 which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.

In this paper we give some characterizations of topological extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative topological left invariant mean then T is topological left lumpy if and only if there is a multiplicative topological left invariant mean M on M(S)* such that M(XT)=1, where XT is the characteristic functional of T. Consequently if T is a topological left lumpy locally compact Borel subsemigroup of a locally compact semigroup S, then T is extremely topological left amenable if and only if S is.

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.