Exercise 1. Let \(X\) be a space. Let \(\mathcal{D}\) be a collection of subsets of \(X\) that is maximal with respect to the finite intersection property (FIP). (a) Show that \(x \in \bar{D}\) for every \(D \in \mathcal{D}\) if and only if every nei…

Theorem 18.4 in James Munkres “Topology” states that if a function \(f : A \rightarrow X \times Y\) is continuous, its coordinate functions \(f_1 : A \rightarrow X\) and \(f_2 : A \rightarrow Y\) are also continuous, and the converse is also true. Th…

Exercise 22.6 Recall that \(\mathbb{R}_{K}\) denotes the real line in the \(K\)-topology. Let \(Y\) be the quotient space obtained from \(\mathbb{R}_K\) by collapsing the set \(K\) to a point; let \(p: \mathbb{R}_K \rightarrow Y\) be the quotient map…

Exercise 22.3 Let \(\pi_1: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be projection on the first coordinate. Let \(A\) be the subspace of \(\mathbb{R}\times\mathbb{R}\) consisting of all points \(x \times y\) for which either \(x \geq 0\)…

Example 1 Let \(X\) be the subspace \([0,1]\cup[2,3]\) of \(\mathbb{R}\), and let \(Y\) be the subspace \([0,2]\) of \(\mathbb{R}\). The map \(p: X \rightarrow Y\) defined by \[ p(x)=\begin{cases} x & \text{for}\; x \in [0,1],\\ x-1 & \text{for}\;…

Lemma 21.2 (The sequence lemma) Let \(X\) be a topological space; let \(A \subset X\). If there is a sequence of points of \(A\) converging to \(x\), then \(x \in \bar{A}\); the converse holds if \(X\) is metrizable. Proof a) Sequence convergence \(\…

Proof of Theorem 20.3 Theorem 20.3 The topologies on \(\mathbb{R}^n\) induced by the euclidean metric \(d\) and the square metric \(\rho\) are the same as the product topology on \(\mathbb{R}^n\). Proof: a) Prove the two metrics can mutually limit ea…

Theorem 20.4 The uniform topology on \(\mathbb{R}^J\) is finer than the product topology and coarser than the box topology; these three topologies are all different if \(J\) is infinite. Proof: a) Prove the uniform topology is finer than the product…

Theorem 19.6 Let \(f: A \rightarrow \prod_{\alpha \in J} X_{\alpha}\) be given by the equation \[ f(a) = (f_{\alpha}(a))_{\alpha \in J}, \] where \(f_{\alpha}: A \rightarrow X_{\alpha}\) for each \(\alpha\). Let \(\prod X_{\alpha}\) have the product…

Theorem 16.3 If \(A\) is a subspace of \(X\) and \(B\) is a subspace of \(Y\), then the product topology on \(A \times B\) is the same as the topology \(A \times B\) inherits as a subspace of \(X \times Y\). Comment: To prove the identity of two topo…