Homework 12: Homeomorphisms

1. Section 4.2, #4.27. 4.30, 4.32(a).
2. Let \(f:A\to B\) be a map of sets. Prove that \(f\) is bijective if and only if there exists a map \(g:B\to A\) such that \(g\circ f=\operatorname{id}_A\) and \(f\circ g=\operatorname{id}_B\).
3. Give an explicit homeomorphism \(f:\mathring{B}^2\to \bR^2\), and give its inverse function explicitly (see top of page 17 for the definition of \(\mathring{B}^2\)). You need not prove that \(f\) or its inverse function are continuous. However, you must explain why both \(f\) and its inverse function are well defined, and verify that they are inverse to eachother. This problem shows that ``boundedness'' is not a topological property.
4. Let \(X\) and \(Y\) be topological spaces. Let \(y_0\in Y\), and let \(f:X\to Y\) be continuous.
   1. Prove that the map \(g:X\to X\times Y\) given by \(g(x)=(x,y_0)\) is an embedding.
   2. Prove that the map \(h:X\to X\times Y\) given by \(h(x)=(x,f(x))\) is an embedding. (Note that \(f\) need not be injective. Note also that the image of \(h\) is the graph of \(f\), as defined in Exercise 4.10 of Section 4.1.)
   Hint. You may find it helpful to use the theorem from class giving rules for constructing continuous functions.