homeomorphism between Boolean spaces

In this entry, we derive a test for deciding when a bijection between two Boolean spaces is a homeomorphism.

We start with two general remarks.

Lemma 1.

If Y is zero-dimensional, then f:XY is continuousMathworldPlanetmathPlanetmath provided that f-1(U) is open for every clopen set U in Y.


Since Y is zero-dimensional, Y has a basis of clopen sets. To check the continuity of f, it is enough to check that f-1(U) is open for each member of the basis, which is true by assumptionPlanetmathPlanetmath. Hence f is continuous. ∎

Lemma 2.

If X is compactPlanetmathPlanetmath and Y is HausdorffPlanetmathPlanetmath, and f is a bijection, then f is a homeomorphism iff it is continuous.


One direction is obvious. We want to show that f-1 is continuous, or equivalently, for any closed setPlanetmathPlanetmath U in X, f(U) is closed in Y. Since X is compact, U is compact, and therefore f(U) is compact since f is continuous. But Y is Hausdorff, so f(U) is closed. ∎

Proposition 1.

If X,Y are Boolean spaces, then a bijection f:XY is homeomorphism iff it maps clopen sets to clopen sets.


Once more, one direction is clear. Now, suppose f maps clopen sets to clopen sets. Since X is zero-dimensional, f-1:YX is continuous by the first propositionPlanetmathPlanetmathPlanetmath. Since Y is compact and X Hausdorff, f-1 is a homeomorphism by the second proposition. ∎

Title homeomorphism between Boolean spaces
Canonical name HomeomorphismBetweenBooleanSpaces
Date of creation 2013-03-22 19:09:04
Last modified on 2013-03-22 19:09:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Result
Classification msc 06E15
Classification msc 06B30
Related topic DualOfStoneRepresentationTheorem