Two easy examples of zero-dimensional spaces

Abstract

ABSTRACT.We present simple examples of a realcompact zerodimensional space which is not Ar-compact and an N-compact space which is not strongly zerodimensional.

The classical ring of quotients of $C_c(X)$

“…Remark 3.10. Let X * denote the finer topology on R 2 defined by Mysior [11] using D = Q × Q. X * is another example of a zero-dimensional space that is not strongly zero-dimensional. The countable set D is precisely the set of all isolated points of X * .…”

Constructing Banaschewski compactification without Dedekind completeness axiom

“…Thus it follows that K(X, F ) is exactly a singleton if and only if either F = R or X is strongly zero dimensional, and in all other cases it has at least two elements. No cardinal estimates for K(X, F ) could be ascertained, and none of the known examples of zerodimensional but not strongly zero-dimensional spaces (see [9,21,22,23,25,26]) could be classified in the exhibited three classes.…”

“…Question 4.4. To which of the classes of zero-dimensional but not strongly zerodimensional spaces discussed above do the spaces discussed in [9,21,22,25] Algebra…”