ThmDex – An index of mathematical definitions, results, and conjectures.

Result R233 on D98: Closed set

Whole space is closed

Formulation 0

Let $T = (X, \mathcal{T})$ be a D1106: Topological space.

Then $X$ is a D98: Closed set in $T$.

Proofs

Proof 0

Let $T = (X, \mathcal{T})$ be a D1106: Topological space.

By definition, the D13: Empty set $\emptyset$ is open in $T$. Result R2066: Difference of set with itself now yields \begin{equation} X \setminus X = \emptyset \in \mathcal{T} \end{equation} The claim follows. $\square$