Zero Dimensional Space is T3
Theorem
Let $T = \struct {S, \tau}$ be a zero dimensional topological space.
Then $T$ is a $T_3$ space.
Proof
Let $T = \struct {S, \tau}$ be a zero dimensional space.
Let $F \subseteq S$ be closed in $T$.
Let also $y \notin F$.
Then by definition of closed, $\relcomp S F$ is open in $T$, where $\relcomp S F$ is the complement of $F$ in $S$.
As $T$ is zero dimensional, it has a basis $\BB$ which consists entirely of clopen sets.
As $\BB$ is a basis for $T$, it follows that:
- $\ds \exists \UU \subseteq \BB: \relcomp S F = \bigcup \UU$
that is, $\relcomp S F$ is the union of a subset of elements of $\BB$.
Thus, there is a set $U \in \UU$ such that $y \in U$ and $F \subseteq \relcomp S U$.
But the elements of $\UU$ are clopen sets, so $U$ is itself clopen.
Thus, by definition, $\relcomp S U$ is also clopen.
So we have that $U$ and $\relcomp S U$ are open sets in $T$ such that:
- $\exists W, R \in \tau: F \subseteq W, y \in R: R \cap W = \O$
by setting $R = U$ and $W = \relcomp S U$.
That is, $T$ is a $T_3$ space.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness