# Quotient Topology

1. Aug 28, 2011

### Tedjn

1. The problem statement, all variables and given/known data

This is from Lee's Introduction to Smooth Manifolds. Suppose π : X → Y is a quotient map. Prove that the restriction of π to any saturated open or closed subset of X is a quotient map.

2. Relevant equations

Lee defines a subset U of X to be saturated if U = π-1(π(U)). π is a quotient map if it is surjective and continuous w.r.t the quotient topology defined by π.

3. The attempt at a solution

My interpretation is that I should prove that, if S is a saturated open or closed subset of X, then π|S is a quotient map between S and π(S) = π|S(S), both spaces being endowed with the subspace topology. That is, show that the quotient topology defined by π|S is equivalent to the subspace topology. Is this correct?

I am imagining this approach in my mind, and I don't see how to use the hypothesis that S is either open or closed rather than any arbitrary subset. This worries me. Any advice?