Limit Point of Subset is Limit Point of Set

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:

$B \cap \paren {U \setminus \set x} \ne \O$

$\blacksquare$