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Homehomeomorphism between Boolean spaces

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# 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 $f:X\to Y$ is continuous provided that $f^{{-1}}(U)$ is open for every clopen set $U$ in $Y$.

###### Proof.

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 assumption. Hence $f$ is continuous. ∎

###### Lemma 2.

If $X$ is compact and $Y$ is Hausdorff, and $f$ is a bijection, then $f$ is a homeomorphism iff it is continuous.

###### Proof.

One direction is obvious. We want to show that $f^{{-1}}$ is continuous, or equivalently, for any closed set $U$ 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 $f:X\to Y$ is homeomorphism iff it maps clopen sets to clopen sets.

###### Proof.

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

## Mathematics Subject Classification

06E15*no label found*06B30

*no label found*

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