Dense Subspace of Extremally Disconnected Space is Extremally Disconnected

Question

Problem 15G of Willard is -

Every dense subspace and every open subspace of an extremally disconnected space is extremally disconnected.

I've been able to prove the 'open subspace' part of the problem, but the result for 'dense subspace' has been evading me. Any help would be appreciated!

Answer 1

$\newcommand{\cl}{\operatorname{cl}}$ Suppose that $D$ is dense in an extremally disconnected space $X$, and let $U$ be (relatively) open in $D$; then $U=D\cap V$ for some open $V$ in $X$. I claim that $\cl_DU=D\cap\cl_XV$. If this is true, then $\cl_DU$ is clearly open in $D$, and the desired result follows.

Proof of claim: Clearly $\cl_XV$ is a closed set in $X$ containing $U$, so $D\cap\cl_XV$ is a closed set in $D$ containing $U$, and therefore $\cl_DU\subseteq D\cap\cl_XV$.

Now suppose that $x\in D\cap\cl_XV$, and let $W$ be any open nbhd of $x$ in $D$; $W=D\cap G$ for some open $G$ in $X$. Clearly $G\cap V\ne\varnothing$, so $$W\cap U=(D\cap G)\cap(D\cap V)=D\cap(G\cap V)\ne\varnothing\;,$$ since $D$ is dense in $X$. Thus, $x\in\cl_DU$, and since $x$ was arbitrary, $D\cap\cl_XV\subseteq\cl_DU$ and hence $\cl_DU=D\cap\cl_XV$. $\dashv$