Limit Points in Particular Point Space
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Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.
Let $x \in S$ such that $x \ne p$.
Then $x$ is a limit point of $p$.
Limit Points in Subset
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
Let $U \in \tau$ be an open set of $T$.
Then by definition of $T$:
- $p \in U$
That is, $U$ contains (at least) one point of $S$ which is distinct from $x$.
As $U$ is arbitrary, it follows that all open set of $T$ have this property.
The result follows by definition of the limit point of a point.
$\blacksquare$
Proof 2
Follows directly from:
$\blacksquare$
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: $2$