On the category of profinite spaces as a reflective subcategory

Abolfazl Tarizadeh

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 the quotient topology 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 $X$.

Keywords

profinite spaces; connected components; coarser topology; reflective subcategory

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References

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Universitat Politècnica de València

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