Problem 21
If \(X\) is an infinite set with the cofinite topology, then every \(f \in C(X)\) is constant.
Problem 28
Let \(X\) be a topological space equipped with an equivalence relation, \(\tilde{X}\) the set of equivalence classes, \(\pi: X \rightarrow \widetilde{X}\) the map taking each \(x \in X\) to its equivalence class, and \(\mathcal{T}=\left\\{U \subset \widetilde{X}: \pi^{-1}(U)\right.\) is open in \(\left.X\right\\} .\) a. \(\mathcal{J}\) is a topology on \(\widetilde{X}\). (It is called the quotient topology.) b. If \(Y\) is a topological space, \(f: \widetilde{X} \rightarrow Y\) is continuous iff \(f \circ \pi\) is continuous. c. \(\widetilde{X}\) is \(T_{1}\) iff every equivalence class is closed.
Problem 30
If \(A\) is a directed set, a subset \(B\) of \(A\) is called cofinal in \(A\) if for each \(\alpha \in A\) there exists \(\beta \in B\) such that \(\beta \gtrsim \alpha\). a. If \(B\) is cofinal in \(A\) and \(\left\langle x_{\alpha}\right\rangle_{\alpha \in A}\) is a net, the inclusion map \(B \rightarrow A\) makes \(\left\langle x_{\beta}\right\rangle_{\beta \in B}\) a subnet of \(\left\langle x_{\alpha}\right\rangle_{\alpha \in A}\). b. If \(\left\langle x_{\alpha}\right\rangle_{\alpha} \in A\) is a net in a topological space, then \(\left\langle x_{\alpha}\right\rangle\) converges to \(x\) iff for every cofinal \(B \subset A\) there is a cofinal \(C \subset B\) such that \(\left\langle x_{\gamma}\right\rangle_{\gamma \in C}\) converges to \(x\).
Problem 34
If \(X\) has the weak topology generated by a family \(F\) of functions, then \(\left\langle x_{\alpha}\right\rangle\) converges to \(x \in X\) iff \(\left\langle f\left(x_{\alpha}\right)\right\rangle\) converges to \(f(x)\) for all \(f \in \mathscr{F}\). (In particular, if \(X=\prod_{i \in I} X_{i}\), then \(x_{\alpha} \rightarrow x\) iff \(\pi_{i}\left(x_{\alpha}\right) \rightarrow \pi_{i}(x)\) for all \(\left.i \in I .\right)\)
Problem 37
Let \(0^{\prime}\) denote a point that is is not an element of \((-1,1)\), and let
\(X=(-1,1) \cup\) \(\left\\{0^{\prime}\right\\}\). Let \(\mathcal{T}\) be the
topology on \(X\) generated by the sets \((-1, a),(a, 1),[(-1, b) \backslash\)
\(\\{0\\}] \cup\left\\{0^{\prime}\right\\}\), and \([(c, 1) \backslash\\{0\\}]
\cup\left\\{0^{\prime}\right\\}\) where \(-1
Problem 38
Suppose that \((X, \mathcal{J})\) is a compact Hausdorff space and \(\mathcal{T}^{\prime}\) is another topology on \(X .\) If \({ }^{\prime}{ }^{\prime}\) is strictly stronger than \({\mathcal{J}}\), then \(\left(X, \mathcal{J}^{\prime}\right)\) is Hausdorff but not compact. If \(\mathcal{J}^{\prime}\) is strictly weaker than \(\mathcal{J}\), then \(\left(X, \mathcal{T}^{\prime}\right)\) is compact but not Hausdorff.
Problem 51
If \(X\) and \(Y\) are topological spaces, \(\phi \in C(X, Y)\) is called proper if \(\phi^{-1}(K)\) is compact in \(X\) for every compact \(K \subset Y\). Suppose that \(X\) and \(Y\) are LCH spaces and \(X^{*}\) and \(Y^{*}\) are their one-point compactifications. If \(\phi \in C(X, Y)\), then \(\phi\) is proper iff \(\phi\) extends continuously to a map from \(X^{*}\) to \(Y^{*}\) by setting \(\phi(\infty, X)=\infty_{Y}\).
Problem 52
The one-point compactification of \(\mathbb{R}^{n}\) is homeomorphic to the \(n\)-sphere \(\\{x \in\) \(\left.\mathbb{R}^{n+1}:|x|=1\right\\}\).
Problem 54
Let \(Q\) have the relative topology induced from \(R\). a. \(\mathbb{Q}\) is not locally compact. b. \(\mathbb{Q}\) is \(\sigma\)-compact (it is a countable union of singleton sets), but uniform convergence on singletons (i.e., pointwise convergence) does not imply uniform convergence on compact subsets of \(Q\).
Problem 57
An open cover \(\mathcal{U}\) of a topological space \(X\) is called locally finite if each \(x \in X\) has a neighborhood that intersects only finitely many members of \(U\). If \(U, \mathcal{V}\) are open covers of \(X, V\) is a refinement of \(U\) if for each \(V \in V\) there exists \(U \in \mathcal{U}\) with \(V \subset U . X\) is called paracompact if every open cover of \(X\) has a locally finite refinement. a. If \(X\) is a \(\sigma\)-compact LCH space, then \(X\) is paracompact. In fact, every open cover \(U\) has locally finite refinements \(\left\\{V_{\alpha}\right\\},\left\\{W_{\alpha}\right\\}\) such that \(\bar{V}_{\alpha}\) is compact and \(\bar{W}_{\alpha} \subset V_{\alpha}\) for all \(\alpha\). (Let \(\left\\{U_{n}\right\\}_{1}^{\infty}\) be as in Proposition 4.39. For each \(n\), \(\left\\{E \cap\left(U_{n+2} \backslash \bar{U}_{n-1}\right): E \subset U\right\\}\) is an open cover of \(\bar{U}_{n+1} \backslash U_{n}\). Choose a finite
