Element of Topological Space may or may not be Limit Point

Theorem

Let $S$ be a set.

Let $H \subseteq S$ be a subset of $S$.

Let $T = \struct {H, \tau}$ be a topological space on the underlying set $H$.

Let $a \in H$.

Then $a$ may or may not be a limit point of $T$.

Whether it is or not depends upon the nature of both $a$ and $T$.

Proof

Let $\R$ be the set of real numbers.

Let $H \subseteq \R$ be the subset of $\R$ defined as:

$H = \set 0 \cup \openint 1 2$

From Limit Point Examples: Union of Singleton with Open Real Interval, $0$ is not a limit point of $H$, although $0 \in H$.

From Limit Point Examples: End Points of Real Interval, $a$ is a limit point of $\closedint a b$ where $a \in \closedint a b$.

Hence the result.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.10 \ \text {(b)}$