Set of Condensation Points is Subset of Derivative

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A$ be a subset of $S$.


Then:

$A^0 \subseteq A'$

where

$A^0$ denotes the set of condensation points of $A$
$A'$ denotes the derivative of $A$


Proof

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 $A$

By definition of derived set:

$x \in A'$

$\blacksquare$


Sources