ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1275 on D55: Continuous map
Every map to trivial topological space is continuous
Formulation 0
Let $T_X = (X, \mathcal{T}_X)$ be a D1106: Topological space.
Let $T_Y = (Y, \{ \emptyset, Y \})$ be a D1162: Bottom 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 $T_X = (X, \mathcal{T}_X)$ be a D1106: Topological space.
Let $T_Y = (Y, \{ \emptyset, Y \})$ be a D1162: Bottom topological space.
Let $f : X \to Y$ be a D18: Map from $X$ to $Y$.
We have $f^{-1} \emptyset = \emptyset$ which is open in $T_X$ by definition. Next, by definition, a map is a D359: Left-total binary relation and therefore \begin{equation} f^{-1} Y = \{ x \in X : f(x) \in Y \} = X \end{equation} which is again open in $X$ by definition. The claim now follows from result R324: Continuity characterised by preimages of open sets. $\square$