Equivalence of Definitions of Points Separated by Neighborhoods

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Theorem

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


The following definitions of the concept of Points Separated by Neighborhoods are equivalent:

Definition 1

Let $x, y \in S$ such that:

$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \in U \subseteq N_x, y \in V \subseteq N_y: N_x \cap N_y = \O$


That is, that $x$ and $y$ both have neighborhoods in $T$ which are disjoint.


Then $x$ and $y$ are described as separated by neighborhoods.

Definition 2

Let $x, y \in S$ such that:

$\exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$


That is, that $x$ and $y$ both have open neighborhoods in $T$ which are disjoint.


Then $x$ and $y$ are described as separated by (open) neighborhoods.


Proof

Let $x, y \in S$.


From Singleton of Element is Subset:

$x$ and $y$ are separated as points by neighborhoods if and only if the singletons $\set x$ and $\set y$ are separated as sets by neighborhoods.


From Equivalence of Definitions of Sets Separated by Neighborhoods:

the singletons $\set x$ and $\set y$ are separated as sets by neighborhoods if and only if the singletons $\set x$ and $\set y$ are separated as sets by open sets.


From Singleton of Element is Subset:

the singletons $\set x$ and $\set y$ are separated as sets by open sets if and only if $x$ and $y$ are separated as points by open sets.

$\blacksquare$