Equivalence of Definitions of T1 Space

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Theorem

The following definitions of the concept of $T_1$ (Fréchet) space are equivalent:


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

Definition by Open Sets

$\left({S, \tau}\right)$ is a Fréchet space or $T_1$ space if and only if:

$\forall x, y \in S$ such that $x \ne y$, both:
$\exists U \in \tau: x \in U, y \notin U$
and:
$\exists V \in \tau: y \in V, x \notin V$

Definition by Closed Points

$\left({S, \tau}\right)$ is a Fréchet space or $T_1$ space if and only if all points of $S$ are closed in $T$.


Proof

Definition by Open Sets implies Definition by Closed Points

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

$\forall x, y \in S$, both:
$(1): \quad \exists U \in \tau: x \in U, y \notin U$

and

$(2): \quad \exists U \in \tau: y \in U, x \notin U$


Let $x, y \in S$.

By the definition of limit point of a point, the above condition means:

$(3): \quad y$ is a limit point of $x$ if every open set $U \in \tau$ such that $y \in U$ contains $x$.


Thus $(2)$ and $(3)$ give us that $y$ is not a limit point of $x$.


As $x$ and $y$ are any two points in $S$, it follows that $x$ has no limit points.

Thus it holds vacuously that all the limit point of $x$ are in $\left\{{x}\right\}$.


By Closed Set iff Contains all its Limit Points, we have that $\left\{{x}\right\}$ is a closed set in $T$.

This holds for all $x \in S$.

Hence $T = \left({S, \tau}\right)$ is a topological space for which all points are closed.

$\Box$


Definition by Closed Points implies Definition by Open Sets

$T = \left({S, \tau}\right)$ is a topological space for which all points are closed.

Let $x, y \in S$.

By Closed Set iff Contains all its Limit Points, we have that:

all the limit points of $x$ are in $\left\{{x}\right\}$
all the limit points of $y$ are in $\left\{{y}\right\}$.

It follows by definition of limit point (reversing the above argument) that:

$\exists U \in \tau: x \in U, y \notin U$

and

$\exists U \in \tau: y \in U, x \notin U$

$\blacksquare$


Sources