Characterization of Derivative by Open Sets

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

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

Let $x$ be a point of $T$.

Then
 * $x \in \operatorname{Der} A$ iff for every open subset $U$ of $T$ if $x \in U$, then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \neq y$

where
 * $\operatorname{Der} A$ denotes the derivative of $A$.

Proof
To prove the first implication assume $x \in \operatorname{Der} A$.

Then $x$ is an accumulation point of $A$ by Definition:Set derivative.

Then (1): $x \in \operatorname{Cl} \left({(A \setminus \{x\}}\right)$ by Definition:Accumulation Point of a Set.

Let $U$ be an open subset of $T$;

Assume $x \in U$.

Then $\left({A \setminus \{x\}}\right) \cap U \neq \varnothing$ by (1), Condition for Point being in Closure

Then there exists $y$ being a point such that
 * (2): $y \in A \setminus \{x\}$ and $y \in U$.

Then $y \in A$ and $y \not\in \{x\}$ by Definition:Set Difference

Hence $y \in A \cap U$ and $x \neq y$ by (2), Definition:Set Intersection, Definition:Singleton

To prove the second implication let us assume

(3): for every open subset $U$ of $T$ if $x \in U$, then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \neq y$

Let us prove that for every open subset $G$ of $T$ if $x \in G$, then $\left({A \setminus \{x\}}\right) \cap U \neq \varnothing$
 * Let $G$ be an open subset of $T$ such that $x \in G$.
 * Then there exists a point $y$ of $T$ such that
 * (4): $y \in A \cap G$ and $x \neq y$ by (3).
 * Then $y \in A$ and $y \not\in \{x\}$ by Definition:Set Intersection, Definition:Singleton.
 * Then $y \in A \setminus \{x\}$ by Definition:Set Difference.
 * $y \in G$ by (4), Definition:Set Intersection.
 * Hence $\left({A \setminus \{x\}}\right) \cap U \neq \varnothing$.

Then $x \in \operatorname{Cl} \left({A \setminus \{x\}}\right)$ by Condition for Point being in Closure.

Then $x$ is an accumulation point of $A$ by Definition:Accumulation Point of a Set.

Hence $x \in \operatorname{Der} A$ by Definition:Set derivative.