Equivalence of Definitions of Limit Point/Definition (1) iff Definition (4)/Proof 1

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

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

A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:

$A \cap \paren {U \setminus \set x} \ne \O$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.

A point $x \in S$ is a limit point of $A$ if and only if $\left({S \setminus A}\right) \cup \left\{{x}\right\}$ is not a neighborhood of $x$.

The following equivalence holds:

$\ds $

$\ds $

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

$\ds $

$\leadstoandfrom$

$\ds $

There exists an open neighborhood $U$ of $x$ such that $U \subseteq \paren{S \setminus A} \cup \set x$

$\ds $

$\leadstoandfrom$

$\ds $

$\paren{S \setminus A} \cup \set x$ is a neighborhood of $x$

$\quad$ Definition of Neighborhood of Point

The result follows from the Rule of Transposition.

$\blacksquare$