Set of Condensation Points is Monotone

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A, B$ be subsets of $S$.

Then:
 * $A \subseteq B \implies {A^0} \subseteq B^0$

where
 * $A^0$ denotes the set of condensation points of $A$

Proof
Assume
 * $A \subseteq B$

Let $x \in A^0$.

By definition of set of condensation points:
 * $x$ is condensation point of $A$

By definition of condensation point:
 * $x$ is limit point of $A$ such that $\forall U \in \tau: A \cap U$ is uncountable

Thus by Limit Point of Subset is Limit Point of Set:
 * $x$ is limit point of $B$

Let $U \in \tau$.

By definition of condensation point:
 * $A \cap U$ is uncountable

By Set Intersection Preserves Subsets/Corollary:
 * $A \cap U \subseteq B \cap U$

Thus by Subset of Countable Set is Countable
 * $B \cap U$ is uncountable

Then by definition:
 * $x$ is condensation point of $B$

Thus by definition of set of condensation points:
 * $x \in B^0$