Homework 9: Quotient Topology

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 3.3, #3.23, 3.24, 3.27, 3.33(a)(b)(c)(h)(j)(k)
2. Let \(X\) and \(S\) be sets, and let \(g:X\to S\) be a surjective map. Define a relation \(\sim\) on \(X\) by \(x\sim y\) if \(g(x)=g(y)\).
   1. Verify that \(\sim\) is an equivalence relation.
   2. What are the equivalence classes of \(\sim\)?
   3. Give a bijection \(\overline{g}:(X/\sim)\to S\). Your bijection \(\overline{g}\) should make the following diagram commute:
      where \(\pi\) is the standard projection. In other words, \(g\) should equal \(\overline{g}\circ \pi\) for your bijection \(\overline{g}\) . We sometimes say that \(g\) factors as the composition of the two maps \(\overline{g}\) and \(\pi\), or, more commonly, that \(g\) factors through \(X/ \sim\).
3. Let \(X\) be a topological space, \(S\) a set, and \(g:X\to S\) a surjective map. Give \(S\) the quotient topology. Let \(C\) be a subset of \(S\). Prove that \(C\) is closed in \(S\) if and only if \(g^{-1}(C)\) is closed in \(X\).