Limit Point of Set may or may not be Element of Set
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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.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.10 \ \text {(a)}$