In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such that $\newcommand{\cl}{\operatorname{cl}} \cl_{X}{A} = \cl_{X}({\bigcup_{\alpha\in S}A_{\alpha}})$.

Also, I defined $X$ to be an $EB$-space if for every cozero-set $H$ of $COZ[X]$, there exists a collection $\{H_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such that $\cl_{X}{H} = \cl_{X}({\bigcup_{\alpha\in S}H_{\alpha}})$.

For example, any zero-dimensional space is an $EZ$-space, and any basically disconnected space is an $EB$-space.

**Problem:** Give an example of a completely regular $EB$-space which is not an $EZ$-space.