On the category of profinite spaces as a reflective subcategory

Abolfazl Tarizadeh

Iran, Islamic Republic of

University of Maragheh

Faculty of Basic Sciences

Submitted: 2013-07-02

|

Accepted: 2013-07-02

|

Published: 2013-07-02

DOI: https://doi.org/10.4995/agt.2013.1575
Funding Data

Downloads

Keywords:

profinite spaces, connected components, coarser topology, reflective subcategory

Supporting agencies:

This research was not funded

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 $X$.
Show more Show less

References:

R. Bkouche, Couples spectraux et faisceaux associés. Applications aux anneaux de fonctions, Bull. Soc. Math. France 98 (1970), 253-295.

F. Borceux and G. Janelidze, Galois Theories, Cambridge University Press, 2001. http://dx.doi.org/10.1017/CBO9780511619939

T. C. Craven, The Boolean space of orderings of a field}, Trans. Amer. Math. Soc. 209 (1975), 225-235. http://dx.doi.org/10.1090/S0002-9947-1975-0379448-X

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976.

T. Szamuely, Galois Groups and Fundamental Groups, Cambridge Studies in Adv. Math., vol 117, 2009. http://dx.doi.org/10.1017/CBO9780511627064

Show more Show less