  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 $X$.
