Set of Condensation Points is Monotone

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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:

$x \in B^0$

$\blacksquare$


Sources