# Equivalence of Definitions of Limit Point

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## Theorem

The following definitions of the concept of **limit point** are equivalent:

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 $x$.

## Proof

### $({1}) \iff ({2})$

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.

$\Box$

### $({2}) \iff ({3})$

Follows directly from Equivalence of Definitions of Adherent Point.

$\Box$

### $({1}) \iff ({4})$

The following equivalence holds:

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

\(\ds \) | \(\leadstoandfrom\) | \(\ds \) | There exists an open neighborhood $U$ of $x$ such that $U \subseteq H^{\complement} \cup \set x$ | \(\quad\) Modus Ponendo Tollens | ||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds \) | $H^{\complement} \cup \set x$ is a neighborhood of $x$ | \(\quad\) Definition of Neighborhood of Point |

The result follows from the Rule of Transposition.

$\blacksquare$

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### $({1}) \iff ({4})$, Proof Variant

The following equivalence holds:

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

\(\ds \O\) | \(=\) | \(\ds H \cap \paren {U \setminus \set x}\) | ||||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \O\) | \(=\) | \(\ds \paren {U \cap H} \setminus \set x\) | \(\quad\) Intersection with Set Difference is Set Difference with Intersection | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \O\) | \(=\) | \(\ds \paren {H \cap U} \setminus \set x\) | \(\quad\) Intersection is Commutative | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \O\) | \(=\) | \(\ds U \cap \paren {H \setminus \set x}\) | \(\quad\) Intersection with Set Difference is Set Difference with Intersection | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \O\) | \(=\) | \(\ds U \cap \map \complement { \map \complement {H \setminus \set x} }\) | \(\quad\) Complement of Complement | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds U\) | \(\subseteq\) | \(\ds \map \complement {H \setminus \set x}\) | \(\quad\) Intersection with Complement is Empty iff Subset | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds U\) | \(\subseteq\) | \(\ds \map \complement {H \cap \map \complement {\set x} }\) | \(\quad\) Set Difference as Intersection with Complement | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds U\) | \(\subseteq\) | \(\ds \map \complement H \cup \map \complement {\map \complement {\set x} }\) | \(\quad\) De Morgan's Laws (Set Theory)/Set Complement | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds U\) | \(\subseteq\) | \(\ds \map \complement H \cup \set x\) | \(\quad\) Complement of Complement |

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

The result follows from the Rule of Transposition.

$\Box$