Equivalence of Definitions of Limit Point

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

Let $H \subseteq S$.

Let $H^{\complement}$ denote the relative complement of $H$ in $S$.

Then the following conditions are equivalent for any point $x \in S$:


 * $(1): \quad$ Every open neighborhood $U$ of $x$ satisfies $H \cap \paren {U \setminus \set x} \ne \O$.


 * $(2): \quad x$ belongs to the closure of $H$ but is not an isolated point of $H$.


 * $(3): \quad x$ is an adherent point of $H$ but is not an isolated point of $H$.


 * $(4): \quad H^{\complement} \cup \set x$ is not a neighborhood of $x$.

$({1}) \iff ({2})$
The closure of $H$ is defined as the union of the set of all isolated points of $H$ and the set of all limit points of $H$.

The rest then follows directly from Equivalence of Definitions of Isolated Point.

$({2}) \iff ({3})$
Follows directly from Equivalence of Definitions of Adherent Point.

$({1}) \iff ({4})$
The following equivalence holds:

The result follows from the Rule of Transposition.

$({1}) \iff ({4})$, Proof Variant
The following equivalence holds:

There exists an open neighborhood $U$ of $x$ such that $H \cap \paren {U \setminus \set x} = \O$

By, $\map \complement H \cup \set x$ is a neighborhood of $x$

The result follows from the Rule of Transposition.

Also see

 * Definition:Limit Point of Set