Singleton Point is Isolated

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Then $x$ is an isolated point of the singleton set $\set x$, but not necessarily an isolated point of $T$.

Proof

Let $U \in \tau$ be an open set of $T$ such that $x \in T$.

The fact that such a $U$ exists follows from the fact that:

from Open Set Axiom $\paren {\text O 3 }$: Underlying Set is Element of Topology, $S$ is open in $T$

$x \in S$.

Hence:

$\set x \subseteq S$

and so from Intersection with Subset is Subset‎: $\set x \cap U = \set x$

So by definition, $x$ is an isolated point of $\set x$.

From Topological Space is Discrete iff All Points are Isolated, unless $T$ is the discrete space on $S$, not all elements of $T$ are isolated points of $T$.

$\blacksquare$