Equivalence of Definitions of T1 Space

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space.

The following two sets of conditions defining a Fréchet ($T_1$) space are logically equivalent:

Definition by Open Sets
$T$ is a Fréchet ($T_1$) space iff:


 * $\forall x, y \in X$, both:
 * $\exists U \in \vartheta: x \in U, y \notin U$
 * and:
 * $\exists V \in \vartheta: y \in V, x \notin V$

That is, for any two points $x, y \in X$ there exist open sets $U, V \in \vartheta$ such that $x$ is in $U$ but not in $V$, and $y$ is in $V$ but not in $U$.

Definition by Closed Points
$T$ is a Fréchet ($T_1$) space iff all points of $T$ are closed.

Definition by Open Sets implies Definition by Closed Points
Let $T = \left({X, \vartheta}\right)$ be a topological space for which:


 * $\forall x, y \in X$, both:
 * $(1): \quad \exists U \in \vartheta: x \in U, y \notin U$

and
 * $(2): \quad \exists U \in \vartheta: y \in U, x \notin U$

Let $x, y \in X$.

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 $U \in \vartheta$ 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 $X$, 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 contains All its Limit Points, we have that $\left\{{x}\right\}$ is a closed set in $T$.

This holds for all $x \in X$.

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

Definition by Closed Points implies Definition by Open Sets
$T = \left({X, \vartheta}\right)$ is a topological space for which all points are closed.

Let $x, y \in X$.

By Closed Set 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 [[Equivalent Definitions for T1 Space#Definition by Open Sets implies Definition by Closed Points|above argument) that:
 * $\exists U \in \vartheta: x \in U, y \notin U$

and
 * $\exists U \in \vartheta: y \in U, x \notin U$