Example of a topological space..... - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:05:56Z http://mathoverflow.net/feeds/question/119262 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119262/example-of-a-topological-space Example of a topological space..... Ali 2013-01-18T12:14:25Z 2013-01-18T12:27:20Z <p>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 $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such that $cl_{X}{A}=cl_{X}({\bigcup_{\alpha\in S}A_{\alpha}})$. </p> <p>Also I defined $X$ to be $EB$-space if 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 $cl_{X}{H}=cl_{X}({\bigcup_{\alpha\in S}H_{\alpha}})$.</p> <p>For example, any zerodimensinal space is an $EZ$-space. and any basically disconnected space is an $EB$-space.</p> <p>Question: Give an example of a completely regular $EB$-space which is not $EZ$-space?</p> http://mathoverflow.net/questions/119262/example-of-a-topological-space/119265#119265 Answer by Nik Weaver for Example of a topological space..... Nik Weaver 2013-01-18T12:27:20Z 2013-01-18T12:27:20Z <p>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.</p>