In my recent research, I defined a topological space $X$ to be $EZ$-space if for every open subset $A$ of $X$ there exists a collection ${A_{\alpha}: \{A_{\alpha}: \alpha\in S}$ S\}$ of clopen subsets of $X$ such that $cl_{X}{A}=cl_{X}({\bigcup_{\alpha\in S}A_{\alpha}})$.
Also I defined $X$ to be $EB$-space if if for every cozero-set $H$, of $COZ[X]$ there exists a collection ${H_{\alpha}: \{H_{\alpha}: \alpha\in S}$ S\}$ of clopen subsets of $X$ such that $cl_{X}{H}=cl_{X}({\bigcup_{\alpha\in S}H_{\alpha}})$.
For example, any zerodimensinal space is an $EZ$-space. and any basically disconnected space is an $EB$-space.
Question: Give an example of a completely regular $EB$-space which is not $EZ$-space?
