# Limit Point of Set may or may not be Element of Set

## 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 $a \in S$ be a limit point of $T$.

Then $a$ may or may not be an element of $H$.

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

## Proof

Consider:

the open real interval $\openint a b$
the closed real interval $\closedint a b$.

Both of these are subsets of the set of real numbers $\R$.

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

But $a \in \closedint a b$ while $a \notin \openint a b$.

Hence the result.

$\blacksquare$