On the category of profinite spaces as a reflective subcategory

TL;DR

This paper proves that the set of connected components of a compact Hausdorff space forms a profinite space and shows that the category of profinite spaces is reflective within compact Hausdorff spaces, with applications to connected components.

Contribution

It establishes a new characterization of connected components in profinite spaces using continuous functions and provides an alternative proof of the reflectiveness of profinite spaces.

Findings

Connected components of compact Hausdorff spaces form profinite spaces.

Category of profinite spaces is a reflective subcategory of compact Hausdorff spaces.

Connected components of certain spaces can be computed in terms of original spaces.

Abstract

In this paper by using the ring of real-valued continuous functions $C(X)$, we prove a theorem in profinite spaces which states that for a compact Hausdorff space $X$, the set of its connected components $X/_{\sim}$ endowed with some topology $\mathscr{T}$ is a profinite space. Then we apply this result to give an alternative proof to the fact that the category of profinite spaces is a reflective subcategory in the category of compact Hausdorff spaces. Finally, under some circumstances on a space $X$, we compute the connected components of the space $t(X)$ in terms of the ones of the space $X$.