Continuity of a function between topological spaces

A function between topological spaces is continuous when the preimage of every open set in the target space is open in the domain space. This is the standard topological definition of continuity.

f : (X, T_{1}) \to (Y, T_{2}) and \forall U \in T_{2} : f^{- 1} (U) \in T_{1}

f:(X,\mathcal{T}_1)\to (Y,\mathcal{T}_2)\quad\text{and}\quad \forall\,U\in \mathcal{T}_2:\quad f^{-1}(U)\in \mathcal{T}_1

What the symbols mean

$f$	the function mapping from space X to space Y
$X$	the domain set
$Y$	the codomain set
$T_{1}$	the topology on X
$T_{2}$	the topology on Y
$U$	an open set in Y
$f^{- 1} (U)$	the preimage of U under f

About this formula

FieldMathematics
TopicTopology
LevelUniversity
Also calledcontinuous function, continuous map, topological continuity, preimage of open set is open, continuous function definition, continuous mapping topology
SourceWikidata Q170058

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