Example of a completely regular spaces

A topological space $X$ is an $EF$-space if if for any two collections $\mathcal{U}$ and $\mathcal{V}$ of clopen subsets of $X$ with $\bigcup \mathcal{U}\cap \bigcup \mathcal{V}=\emptyset$, we have $\bigcup \mathcal{U}$ and $\bigcup \mathcal{V}$ are completely separated.

By deffinition, it is essy to see the following. $(a)$ If $X$ is an $EF$-space and $\mathcal{U}$ and $\mathcal{V}$ are to collection of clopen subsets of $X$ with $\bigcup \mathcal{U}\cap \bigcup \mathcal{V}=\emptyset$, then $cl(\bigcup \mathcal{U})\cap cl(\bigcup \mathcal{V})=\emptyset.$

In Normal space the converse of $(a)$ is true.

Question: For a completely regular space $X$ is the convesre of $(a)$ true?

• Dear Hart, Yes in Zero-dimensional space the following are equivalent. (1) $X$ ia an EF-spac. (2) $X$ is an extremally disconnected. (3) Any union of clopen subsets of $X$ is $C^{*}$-embedded. (4) If $\mathcal{U}$ and $\mathcal{V}$ are two collections of clopen subsets of $X$ with $\bigcup \mathcal{U}\cap \bigcup \mathcal{V}=\emptyset$, then $cl(\bigcup \mathcal{U})\cap cl(\bigcup \mathcal{V})=\emptyset$. – Ali Jan 26 '13 at 15:13