Equivalence of Definitions of Generic Point of Topological Space

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Theorem

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

Let $x \in S$ be an element of $S$.

The following definitions of the concept of Generic Point of Topological Space are equivalent:

Definition 1

The point $x$ is a generic point of $T$ if and only if the closure of the singleton $\set x$ is $S$.

Definition 2

The point $x$ is a generic point of $T$ if and only if $x$ is contained in every non-empty open subset of $T$.

Proof

Let $\set x^-$ denote the closure of $\set x$ in $T$.

Let $\map \UU x$ denote the system of open neighborhoods of $x$.

We have:

\(\ds \)

\(\)

\(\ds \set x^- = S\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds S \setminus \set x^- = \O\)

Set Difference with Superset is Empty Set

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \bigcup \set{U \in \tau : U \notin \map \UU x} = \O\)

Union of Open Sets Not in System of Open Neighborhoods is Complement of Singleton Closure

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \set{U \in \tau : U \notin \map \UU x} = \set {U \in \tau : U = \O}\)

Set is Subset of Union and Subset of Empty Set iff Empty

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \forall U \in \tau : U \ne \O \implies U \in \map \UU x\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds \forall U \in \tau : U \ne \O \implies x \in U\)

Definition of System of Open Neighborhoods

The result follows.

$\blacksquare$