Homework 15: Components and Total Disconnectedness

1. Let \(X\) be a topological space. Suppose that \(A\) is an open and closed subset of \(X\). Prove that \(A\) is the union of components of \(X\).
2. In this problem we prove that \(\bR\) is not homeomorphic to \(\bR_l\).
   1. Give all continuous functions \(f:\bR\to \bR_l\). Prove your answer. (Hint: Consider the components of \(\bR\) and \(\bR_l\).)
   2. Use (a) to prove \(\bR\not\cong \bR_l\).
3. Let \(X\) be a topological space. Define an equivalence relation on \(X\) by \(x\sim y\) if there exists a connected subset of \(X\) containing both \(x\) and \(y\). In class we verified that \(\sim\) is an equivalence relation and defined the components of \(X\) to be the equivalence classes. Prove that \(X/ \sim\) with quotient topology is totally disconnected.
4. Prove that the complement of a dense open subset of \([0,1]\) is totally disconnected.
5. 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.