I am trying to solve Exercise 3.59 from Introduction to Topological Manifolds, by John Lee.
Let $q: X \rightarrow Y$ be any map and $U \subseteq X$.
Given "If $x\in U$, then every point $x' \in X$ such that $q(x) = q(x')$ is also in $U$. ", I need to show $U$ is $q$-saturated.
My attempt: Since we always have $U \subseteq q^{-1}(q(U))$, we only need show that $q^{-1}(q(U)) \subseteq U$ in this case.
I don't know how to utilize the given condition... Any suggestions? Thanks in Advance!
That seems like a good approach. I would try it by contradiction: suppose $q^{-1}(qU) \subseteq U$ does not hold. What would that mean? That would give you a point, call it $x' \in X$, which is in $q^{-1}(qU)$ but is not in $U$. Then argue that there must be some $x \in U$ such that $q(x) = q(x')$.