Limit Point
Suppose that $$A$$ is a subset of a topological space $$X$$, then a point $$x \in X$$ is called a limit point of $$A$$ when every neighborhood of $$x$$ intersects $$A$$ in some point other than $$x$$ itself
Limit Point iff element of Closure minus a Point
$$x$$ is a limit point iff $$x \in \overline { A \setminus \left\{ x \right\} }$$
$$x$$ is a limit point iff every neighborhood of $$x$$ intersects $$A$$ in some point other than $$x$$ itself, which is equivalent to every neighborhood intersecting $$A \setminus \left\{ x \right\}$$ iff $$x \in \overline { A \\ \left\{ x \right\} }$$.
Limit Points are a Subset of Closure
Suppose that $$A$$ is a subset of a topological space $$X$$, if $$A ^ \prime$$ is the set of limit points of $$A$$, then $$A ^ \prime \subseteq \overline{ A }$$
Suppose that $$x \in A ^ \prime$$, then every neighborhood of $$x$$ intersects $$A$$ therefore $$x \in \overline{ A }$$ as needed.
Closure Equals Limit Points Union Itself
Let $$A$$ be a subset of a topological space $$X$$. If $$A ^\prime$$ is the set of all limit points of $$A$$, then $$\overline{ A } = A^\prime \cup A$$

Since $$A \subseteq \overline{ A }$$ and $$A ^ \prime \subseteq \overline{ A }$$, then $$A ^ \prime \cup A \subseteq \overline{ A }$$

Suppose that $$x \in \overline{ A }$$, if $$x \in A$$, then we would be done, so we assume that $$x \notin A$$ and our goal remains to show that $$x \in A ^ \prime$$, so let $$U$$ be a neighborhood of $$x$$, we want to show that it intersects $$A$$ at a point different than $$x$$, since $$x \in \overline{ A }$$, then we know that $$U$$ intersects $$A$$, since $$x \notin A$$ then $$U$$ cannot intersect $$A$$ at that point so we've shown that $$x \in A ^ \prime$$ as needed.

A Subset of a Topological Space is Closed if and only if it contains all it's Limit Points
Suppose $$A$$ is a subset of a topological space $$X$$ then it is closed if and only if it contains it's limit points
A set is closed iff $$A = \overline{ A }$$ since $$\overline{ A } = A \cup A ^ \prime$$, then this is true iff $$A ^ \prime \subseteq A$$.