Limit Points in Particular Point Space/Subset

Theorem

Let $T = \struct {S, \tau_p}$ be a particular point space.

Let $U \subseteq S$ such that $p \in U$.

Let $x \in S$ such that $x \ne p$.

Then $x$ is a limit point of $U$.

Proof 1

Every open set of $T = \struct {S, \tau_p}$ except $\O$ contains the point $p$ by definition.

So every open set $U \in \tau_p$ such that $x \in U$ contains $p$.

So by definition of the limit point of a set, $x$ is a limit point of $U$.

$\blacksquare$

Proof 2

Follows directly from:

Particular Point Topology is Closed Extension Topology of Discrete Topology

Limit Points in Subset of Closed Extension Space

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $3$