quotient topology

Definition

Let $X$ and $Y$ be topological spaces, let $p: X \to Y$ be a surjective mapping, then the map $p$ is said to be a quotient map provided a subset $U \subseteq Y$ is open in $Y$ iff $p^{-1}(U)$ open in $X$.

If $X$ is a space and $A$ is a set, $p: X \to A$ a surjective map, then there exists exactly one topology $\mathcal{T}$ on $A$ relative to which $p$ is a quotient map, called the quotient topology induced by $p$.

Wherefore, topology $\mathcal{T}$ is defined by consisting of subsets $U \subseteq A$ such that $p^{-1}(U)$ is open in $X$.

Notes

• an open map $f: X \to Y$ is such that for each open set $U$ of $X$, set $f(U)$ open in $Y$
  • c.f. a closed map $f: X \to Y$ is such that for each closed set $U$ of $X$, set $f(U)$ closed in $Y$
  • if $p: X \to Y$ is a surjective continuous map that is either open or closed, then $p$ is a quotient map

References

1. J. R. Munkres, Topology, 2. ed., Pearson new internat. ed. Harlow: Pearson, 2014, pp. 136-138.