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.



Example of a topological space
Not possible, in a completely regular space every open set is a union of cozero sets. Just collect the clopen sets that you use for these subordinate cozero sets.


