A convenient category of topological partial groups

A Fathy

doi:10.1186/s42787-019-0010-4

Abstract

In this paper, the concept of ℘-continuous map is introduced and some of their basic properties are discussed. Also, the category mathbf {K}acute {}, of topological partial groups, as objects and the ℘-morphisms of topological partial groups, as arrows, is introduced, which is alternative to the category K, of topological spaces, as objects and k-continuous maps,as arrows, and satisfies the same nice properties of the category kpg, of underline {k}-partial groups, as objects, and the morphisms of underline {k}-partial groups, as arrows (Abd- Allah et al., J. Egyption Math. Soc 25:276-278, 2017).

Highlights

Abd- Allah et al introduced the concept of topological partial groups and discussed some of their basic properties
Abd- Allah et al introduced the concept of k-partial groups and discussed some of their basic properties
The category Kof topological partial groups, as objects, and the ℘-morphisms of topological partial groups, as arrows, is introduced, which is alternative to the category K, of topological spaces, as objects, and k-continuous maps, as arrows

Summary

Introduction

Definition 13 A subset V of the topological partial group S is called ℘-open if h−1[ V ] is open in C for each a ℘-test map h : C → S Definition 14 A subset A of the topological partial group S is called a ℘-neighbourhood of x ∈ S if there exists a ℘-open set U in S such that x ∈ U ⊆ A. Proposition 1 A subset A ⊆ S of the topological partial group S is a ℘-open set if and only if it is a ℘-neighbourhood of each of its points.

Results

Conclusion