# 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 $f\mathrm{:}X\mathrm{\to}Y$ is continuous^{} provided that ${f}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}U\mathrm{)}$ 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.
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###### Lemma 2.

###### 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.
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###### Proposition 1.

If $X\mathrm{,}Y$ are Boolean spaces, then a bijection $f\mathrm{:}X\mathrm{\to}Y$ 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.
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Title | homeomorphism between Boolean spaces |
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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 |