Homework 2: Basis for a Topology

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

1. Section 1.1, #1.4.
2. Section 1.2, #1.11(a), (b).
3. Let \(\cB\) and \(\cB'\) be two bases on a set \(X\), and let \(\ct_{\!\cB}\) and \(\ct_{\!\cB'}\) be the corresponding topologies generated by the bases. Consider the following two statements:
   (i) For every \(B'\in \cB'\) and \(x\in B'\), there exists \(B\in \cB\) such that \(x\in B\subseteq B'\).
   (ii) For every \(B\in \cB\) and \(x\in B\), there exists \(B'\in \cB'\) such that \(x\in B'\subseteq B\).
   
   1. Prove that \(\ct_{\!\cB}\) is finer than \(\ct_{\!\cB'}\) if and only if (i) holds.
   2. Prove that \(\ct_{\!\cB}=\ct_{\!\cB'}\) if and only if (i) and (ii) both hold. (Hint: use part (a).)
   3. Use the Union Lemma to express (i) in terms of unions.
4. Section 1.2, #1.12, 1.13.