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A continuous map $q:(X,T)\to (X’,T’)$ is said to be a quotient map if

  1. $q$ is surjective
  2. defining the equivalence relation in $X$ given by $x\sim y \Longleftrightarrow q(x)=q(y)$, the induced map $\bar q: (\bar X,\bar T) \to (X’,T’)$ (which is bijective and continuous) is a homeomorphism