quality:corruption, linguistic:paraphrasing, linguistic:voice_change, stylistic:tone
Properties

If f: X → Y {\displaystyle f:X\to Y} and g: Y → Z {\displaystyle g:Y\to Z} are continuous, then so is the composition g ∘ f: X → Z. {\displaystyle g\circ f:X\to Z.} If f: X → Y {\displaystyle f:X\to Y} is continuous and

The possible topologies on a fixed set X are partially ordered: a topology τ 1 {\displaystyle \tau _{1}} is said to be coarser than another topology τ 2 {\displaystyle \tau _{2}} (notation: τ 1 ⊆ τ 2 {\displaystyle \tau _{1}\subseteq \tau _{2}}) if every open subset with respect to τ 1 {\displaystyle \tau _{1}} is also open with respect to τ 2. {\displaystyle \tau _{2}.} Then, the identity map id X: (X, τ 2) → (X, τ 1) {\displaystyle \operatorname {id} _{X}:\left(X,\tau _{2}\right)\to \left(X,\tau _{1}\right)} is continuous if and only if τ 1 ⊆ τ 2 {\displaystyle \tau _{1}\subseteq \tau _{2}} (see also comparison of topologies). More generally, a continuous function (X, τ X) → (Y, τ Y) {\displaystyle \left(X,\tau _{X}\right)\to \left(Y,\tau _{Y}\right)} stays continuous if the topology τ Y {\displaystyle \tau _{Y}} is replaced by a coarser topology and/or τ X {\displaystyle \tau _{X}} is replaced by a finer topology.