Empty Set is Closed/Normed Vector Space

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Theorem

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


Then the empty set $\O$ is closed in $M$.


Proof

From Normed Vector Space is Open in Itself, $X$ is open in $M$.

But:

$\O = \relcomp X X$

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

The result follows by definition of closed set.

$\blacksquare$


Sources