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

Result R76 on D98: Closed set

Empty set is closed

Formulation 0

Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
Let $\emptyset$ be the D13: Empty set.

Then $\emptyset$ is a D98: Closed set in $T$.

Proofs

Proof 0

Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
Let $\emptyset$ be the D13: Empty set.

By definition, the whole space $X$ is open in $T$. Result R2067: Subtracting empty set from set now yields \begin{equation} X \setminus \emptyset = X \in \mathcal{T} \end{equation} The claim follows. $\square$