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A topological space $X$ is called relative extremely disconnected if it has a base $B$(for open subsets) such that disjoint elements in B have disjoint closure, i.e, if $C, D$ in $B$ and $C\cap D=\emptyset$, then $clC\cap clD=\emptyset$. Now, does $R$(real nubmer) with usual topology is a relative extremely disconnected space?

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  • $\begingroup$ Ok. Does exist Hausdorff space $X$ which is not relative extremely disconnected space? $\endgroup$
    – Ali
    Commented Nov 23, 2012 at 19:46
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    $\begingroup$ Avoid modern sloppiness. Keep the original name "extremally disconnected". Same for "basically disconnected". $\endgroup$
    – Gerald Edgar
    Commented Feb 23, 2014 at 17:54

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Actually real open intervals with rational left end-point and irrational right end-point are a base with that property.

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