T2 Space is T1 Space

Theorem

Let $\struct {S, \tau}$ be a $T_2$ (Hausdorff) space.


Then $\struct {S, \tau}$ is also a $T_1$ (Fréchet) space.


Proof

From the definition of $T_2$ (Hausdorff) space:

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


As $U \cap V = \O$ it follows from the definition of disjoint sets that:

$x \in U \implies x \notin V$
$y \in V \implies y \notin U$


So if $x \in U, y \in V$ then:

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

which is precisely the definition of a $T_1$ (Fréchet) space.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms
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