Empty Set is Closed/Topological Space

From ProofWiki
Jump to navigation Jump to search


Let $T = \struct {S, \tau}$ be a topological space.

Then $\O$ is closed in $T$.


From the definition of closed set, $U$ is open in $T = \struct {S, \tau}$ if and only if $S \setminus U$ is closed in $T$.

By definition of topological space, $S$ is open in $T$.

From Set Difference with Self is Empty Set:

$S \setminus S = \O$

Hence $\O$ is closed in $T$.