The aim of this paper is to introduce semitopological $\delta$-group and topological with the concept which arise from approximately algebraic structures. Furthermore, it shown that product space determined $\delta$-topological subspaces a space. Fundamental system open $\delta$-neighborhoods related properties were investigated.

Answering recent question of A.V. Arhangel’skii we construct in ZFC an extremally disconnected semitopological group with continuous inverse having no open Abelian subgroups.

We analyze the numerical representability of total preorders defined on semitopological real algebras through continuous order-preserving real-valued functions that are also additive and multiplicative. The results obtained are used to interpret important concepts arising in Social Choice theory.

For a compact Hausdorff space X , the space SC(X ×X) of separately continuous complex valued functions on X can be viewed as a C∗-subalgebra of C(X)∗∗⊗C(X)∗∗, namely those elements which slice into C(X). The analogous definition for a non-commutative C∗-algebra does not necessarily give an algebra, but we show that there is always a greatest C∗-subalgebra. This thus gives a noncommutative notio...