Equivalence of Definitions of Hausdorff Topological Vector Space

Theorem

Let $K$ be a topological field.

Let $\struct {X, \tau}$ be a topological vector space over $K$.

The following definitions of the concept of Hausdorff Topological Vector Space are equivalent:

Definition 1

We say that $\struct {X, \tau}$ is a Hausdorff topological vector space if and only if it is Hausdorff as a topological space.

Definition 2

We say that $\struct {X, \tau}$ is a Hausdorff topological vector space if and only if:

for each $x \in X$, the singleton $\set x$ is closed in $\struct {X, \tau}$.

Proof

From Topological Vector Space is Hausdorff iff T1, $\struct {X, \tau}$ is Hausdorff if and only if it is $T_1$ space.

From Definition 2 of a $T_1$ space, it follows that $\struct {X, \tau}$ is Hausdorff if and only if:

for each $x \in X$, $\set x$ is closed in $\struct {X, \tau}$.

$\blacksquare$