Extremally Disconnected Metric Space is Discrete

Theorem

Let $M = \left({A, d}\right)$ be a metric space which is extremally disconnected.

Then $M$ is the discrete topology.

Proof

Let $M = \left({A, d}\right)$ be extremally disconnected.

Let $p \in X$.

As $M$ is a metric space, $\left\{{p}\right\}$ can be expressed as:

$\displaystyle \left\{{p}\right\} = \bigcap_{n \in \N^*} \left({N_{1/n} \left({p}\right)}\right)^-$

where:

$N_{1/n} \left({p}\right)$ denotes the open $1/n$-ball neighborhood of $p$

$\left({N_{1/n} \left({p}\right)}\right)^-$ denotes the closure of $N_{1/n} \left({p}\right)$

That is, as the intersection of the closures of the open $1/n$-ball neighborhoods of $p$ for all non-zero natural numbers.

Now let:

$\displaystyle U = \bigcup_{n \in \N^*} N_{1/2n} \left({p}\right) \setminus \left({N_{1/(2n+1)} \left({p}\right)}\right)^-$

Then either $U$ or the complementary set of annuli is an open set which has $p$ as a non-interior limit point provided $\left\{{p}\right\}$ is not open.

So if $M$ is not the discrete topology, it cannot be extremally disconnected.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$