Limit Point of Subset is Limit Point of Set
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B$ be subset of $S$ such that
- $A \subseteq B$
Let $x$ be a point of $S$.
Then:
- if $x$ is limit point of $A$, then $x$ is limit point of $B$.
Proof
Assume $x$ is limit point of $A$.
By definition of limit point it suffices to prove
- $\forall U \in \tau: x \in U \implies B \cap \paren {U \setminus \set x} \ne \O$
Let $U \in \tau$ such that
- $x \in U$
By definition of limit point:
- $A \cap \paren {U \setminus \set x} \ne \O$
By Set Intersection Preserves Subsets/Corollary:
- $A \cap \paren {U \setminus \set x} \subseteq B \cap \paren {U \setminus \set x}$
Thus:
- $B \cap \paren {U \setminus \set x} \ne \O$
$\blacksquare$