Open and Closed Sets in Topological Space
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Then $S$ and $\O$ are both both open and closed in $T$.
Proof
From the definition of closed set, $U$ is open in $T$ if and only if $S \setminus U$ is closed in $T$.
From Underlying Set of Topological Space is Clopen, $S$ is both open and closed in $T$.
From Empty Set is Element of Topology, $\O$ is open in $T$.
From Empty Set is Closed in Topological Space, we have that $\O$ is closed in $T$.
$\blacksquare$
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets