Normed Vector Space is Closed in Itself

Theorem
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Then $X$ is closed in $M$.

Proof
From Empty Set is Open in Normed Vector Space, $\O$ is open in $M$.

But:
 * $X = \relcomp X \O$

where $\complement_X$ denotes the set complement relative to $X$.

The result follows by definition of closed set.

Also see

 * Space is Closed in Itself