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

Result R1276 on D55: Continuous map

Every map from discrete topological space is continuous

Formulation 0

Let $T_X = (X, \mathcal{P}(X))$ be a D1110: Discrete topological space.
Let $T_Y = (Y, \mathcal{T}_Y)$ be a D1106: Topological space.
Let $f : X \to Y$ be a D18: Map from $X$ to $Y$.

Then $f$ is a D55: Continuous map from $T_X$ to $T_Y$.

Proofs

Proof 0

Let $T_X = (X, \mathcal{P}(X))$ be a D1110: Discrete topological space.
Let $T_Y = (Y, \mathcal{T}_Y)$ be a D1106: Topological space.
Let $f : X \to Y$ be a D18: Map from $X$ to $Y$.

For all open sets $U \in \mathcal{T}_Y$ in $T_Y$, we have $f^{-1} U \subseteq X$ by definition of a preimage and thus $f^{-1} U \in \mathcal{P}(X)$. The claim therefore is a consequence of result R324: Continuity characterised by preimages of open sets. $\square$