ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
Power set
Hyperpower set sequence
Hyperpower set
Hypersubset
Subset algebra
Subset structure
Topological space
Definition D55
Continuous map
Formulation 0
Let $T_X = (X, \mathcal{T}_X)$ and $T_Y = (Y, \mathcal{T}_Y)$ each be a D1106: Topological space.
A D18: Map $f : X \to Y$ is continuous at $x_0 \in X$ with respect to $T_X$ and $T_Y$ if and only if \begin{equation} \forall \, U \in \mathcal{T}_X \left( x_0 \in U \quad \implies \quad \exists \, V \in \mathcal{T}_Y : U \subseteq f^{-1}(V) \right) \end{equation}
Children
Homeomorphism
Set of continuous maps
Results
Every map from discrete topological space is continuous
Every map to trivial topological space is continuous
Identity map is continuous if topology on domain set is equal or stronger