Problem Sets

If you haven't worked with open sets before or need a refresher, try proving these results. These are all standard results, but please try to prove them without looking them up. Do not turn these in; the first graded problem set will be assigned next week.

When \(X\) is a metric space, we equip it with the metric topology unless otherwise specified.

1. Let \((X,d)\) be a metric space, let \(x\in X\), let \(r>0\). Show that \[B(x,r)=\{y\in X: d(x,y) < r\}\] is an open set.
2. Show that the metric topology on \(X\) is in fact a topology.
3. Give an example of a metric space \(X\) and a sequence of open sets \(U_1,\dots, U_n\) such that \(\bigcap_i U_i\) is not an open set.
4. Let \((X,d)\) be a metric space. Show that every open subset of \(X\) is a union of open balls. (Note that this need not be a countable union of open balls.)
5. Let \((X,\mathscr{S})\), \((Y,\mathscr{T})\) be topological spaces and let \(f\from X\to Y\) be a continuous map. Show that if \(x_n\to a\), then \(f(x_n) \to f(a)\).

Please keep in mind the course's generative AI policy. If you use an LLM in your problem set, you must acknowledge it at the start of your solution to each problem, for instance "I used [LLM] to [edit/generate] [part/most/all] of this solution". At grader discretion, LLM-generated solutions may receive a flat 95% grade (regardless of correctness) and no further feedback.

1. Let \(B_\Q=\{(a,b)\subset \R\mid a,b\in \Q\}\) be the set of open intervals with rational endpoints. Show that \(B_\Q\) is a basis for a topology on \(\R\) and that it generates the standard topology on \(\R\).

2. (based on Munkres, p. 92, #9) Show that the lexicographic order topology on \(\R\times \R\) is equal to the product topology \(\R_d\times \R\), where \(\R_d\) is the discrete topology on \(\R\).

   Conclude that this topology is metrizable (i.e., it is the metric topology for some metric on \(\R\times \R\).

3. We say that a topological space \((X,\mathcal{T})\) is Hausdorff if for every \(x,y\in X\) with \(x\ne y\), there are open sets \(U\) and \(V\) such that \(x\in U\), \(y\in V\), and \(U\cap V = \emptyset\). Show that if \(X\) is finite and Hausdorff, then \(\mathcal{T}\) is the discrete topology.
4. Let \(X = [0,1]^2\), equipped with the lexicographic order topology. Show that every sequence in \(X\) has a convergent subsequence.

Conventions: Unless otherwise specified:

• \(\R^n\) is equipped with the standard topology (equivalently, the metric topology)
• If \(X\) is a topological space and \(Y\subset X\), then \(Y\) is equipped with the subspace topology

1. Show that \(X\) is Hausdorff if and only if the diagonal \[D=\{(x,x)\in X\times X\mid x\in X\}\] of \(X\times X\) is closed.

For 2-4: in class, we stated that if \(X = \bigcup_{i=1}^n K_i\), where the \(K_i\)'s are closed sets, and \(f: X\to Y\) is a function such that \(f|_{K_i}\) is continuous for every \(i\), then \(f\) is continuous.

2. Prove this statement.

3. In general, this fails. For instance, if \(X\) is Hausdorff, \(f:X\to Y\) is not discontinuous, and \(K_x = \{x\}\) for all \(x\in X\), then \(K_x\) is closed for all \(x\), \(X=\bigcup_{x\in X} K_x\), and \(f|_{K_x}\) is continuous for all \(x\), but \(f\) is discontinuous.

   Find countably many closed subsets \(K_1, K_2, \dots \subset \R\) and a function \(f: \R \to \R\) such that \(\R = \bigcup_{i=1}^\infty K_i\) and \(f|_{K_i}\) is continuous for all \(i\), but \(f\) is discontinuous.

4. Suppose that \(X\) is the union of a locally finite collection of closed sets. That is, \(X = \bigcup_{\alpha \in A} K_\alpha\), where \(K_\alpha\) is closed for all \(\alpha\) and where for all \(x\in X\), there is a neighborhood \(U\) of \(x\) such that \(U\) intersects only finitely many of the \(K_\alpha\)'s.

   Show that if \(f: X\to Y\) is a function such that \(f|_{K_\alpha}\) is continuous for all \(\alpha\), then \(f\) is continuous.

5. (Munkres, p. 152, #2) Let \(\{A_n\}\) be a sequence of connected subspaces of \(X\) such that \(A_n\cap A_{n+1}\ne \emptyset\) for all \(n\). Show that \(\bigcup_n A_n\) is connected.
6. Suppose that \(U\subset \R\) is an open set. Show that every connected component of \(U\) is an open interval and show that \(U\) is the disjoint union of at most countably many open intervals.

1. If \(X\) is an ordered set with the order topology and \(Y\subset X\), we can put two topologies on \(Y\): the order topology \(\mathcal{T}_{\mathrm{ord}}\) corresponding to the order inherited from \(X\) and the subspace topology \(\mathcal{T}_{\mathrm{sub}}\) corresponding to the order topology on \(X\).

   Show that these two topologies need not be the same by showing that if \(X=\R\) and \(Y = [0,1) \cup [2,3)\), then \((Y, \mathcal{T}_{\mathrm{ord}})\) is connected, but \((Y, \mathcal{T}_{\mathrm{sub}})\) is not connected.

2. Let \(X\) be a nonempty compact Hausdorff space. Let \(f\from X\to X\) be a continuous map. We write \[f^n(X)=(\underbrace{f\circ f\circ \dots\circ f}_{\text{$n$ times}})(X).\] Show that \(\bigcap_n f^n(X)\) is nonempty. Is the same thing true when \(X\) is noncompact?
3. (Munkres, p. 162, #4) Let \(X\) be locally path-connected. Show that every open connected subset of \(X\) is path-connected.
4. Let \(X\) be a countable ordered set. Show that there is an order-preserving map from \(X\) to \(\R\).

5. Let \(X\) be a nonempty countable well-ordered set and let \(L_X = X\times [0,1)\) with the lexicographic order topology. Show that there is an order-preserving bijection from \(L_X\) to \([0,\infty)\), so that \(L_X\) is homeomorphic to \([0,\infty)\).

   Hint: You can construct an order-preserving map from \(L_X\) to \([0,\infty)\) using techniques similar to #3. To show that it's surjective, you'll need to do some casework. It may help to show the following:

   Suppose that \(X\) is well-ordered and let \(m\) be the smallest element of \(X\). For any \(y\in X\), let \(y + 1\) be the successor of \(y\), i.e., the minimal element of \((y, \infty)\). Then any \(y\in X\) such that \(y\ne m\) is either a successor (there is a \(y'\) such \(y=y' + 1\)) or a limit point (there is an increasing sequence \(y_1,y_2,\dots\) that converges to \(y\)).

6. (Munkres, p. 171, #5) Let \(X\) be Hausdorff and let \(A,B\subset X\) be disjoint compact sets. Show that there exist disjoint open sets \(U,V\) such that \(A\subset U\) and \(B\subset V\).

1. Let \(X\) be a metric space. Let \(K\subset X\) be compact and let \(N\subset X\) be a neighborhood of \(K\). Prove that there is some \(\epsilon > 0\) such that \[\bigcup_{p\in K} B(p,\epsilon) \subset N.\]
2. Let \(X\) be a metric space.
   • For \(n\in \mathbb{N}\), show that there is a metric on \(X^{n}\) that induces the product topology.
   • Show that there is a metric on \(X^{\mathbb{N}}\) that induces the product topology.

3. A metric space \(X\) is totally bounded if for every \(\delta>0\), there is a finite collection of balls of radius \(\delta\) that covers \(X\).

   Show that \(X\) is sequentially compact (and thus compact) if and only if \(X\) is totally bounded and every Cauchy sequence in \(X\) converges.

4. Let \(\mathbb{Q}\) act on \(\R\) by translation. Describe the quotient topology on \(\mathbb{Q}\backslash \R\).
5. Let \(X = \mathbb{N} \times [0,1]\) and let \(\sim\) be the relation \((n,0)\sim (m,0)\) for all \(m,n \in \mathbb{N}\). Let \(Q= X/\sim\).
   • Show that \(Q\) is not compact.
   • Show that \(Q\) is not sequentially compact.
   • Show that \(Q\) is not metrizable.

We are starting on Hatcher - material from class can be found in Chapters 0 and 1 of Hatcher.

There is no class on Monday, so I will hold office hours 1-2pm on Wednesday instead.

1. Suppose that \(X\) is a CW complex. For each cell \(e^n_\beta\) of \(X\), we have maps \[e^n_\beta \hookrightarrow X^{(n-1)} \sqcup \bigsqcup_\alpha e^n_\alpha \xrightarrow{\pi} X^{(n)} \hookrightarrow X.\] The composition is a canonical map \(\chi_\alpha\) from \(e^n_\alpha\) to \(X\), which we call the characteristic map of \(e^n_\alpha\).
   • Let \(f: X\to Y\) be a function. Show that \(f\) is continuous if and only if \(f|_{X^{(n)}}\) is continuous for all \(n\).
   • Let \(f: X\to Y\) be a function. Show that \(f\) is continuous if and only if all of the compositions \(f\circ \chi_\alpha\) are continuous.
2. Show that a CW complex is connected if and only if its 1-skeleton is connected. Make sure to handle the infinite-dimensional case.

3. Construct a manifold \(X\) and a homeomorphism \(\alpha: X\to X\) such that if \(\Gamma\) is the group of homeomorphisms generated by \(\alpha\), then \(\Gamma\) acts on \(X\) freely and discretely, but \(\Gamma\backslash X\) is not Hausdorff.

   (Hint: It's possible to construct such a map with \(X=\R^2\setminus 0\). Note that you must choose \(\alpha\) so that \(\alpha\) does not preserve any metric on \(X\).)

4. Prove that homotopy equivalence is an equivalence relation. (To fix notation, we'll write \(f\sim g\) to say that \(f\) is homotopic to \(g\) and \(X\eqsim Y\) to say that \(X\) and \(Y\) are homotopy equivalent.)

5. Suppose that \(A\subset X\). A retraction onto \(A\) is a map \(r:X\to A\) such that \(r(a)=a\) for all \(a\in A\). A deformation retraction is a retraction \(r:X\to A\) such that \(r\) (considered as a map from \(X\) to \(X\)) is homotopic to \(\id_X\).

   • Show that if there is a deformation retraction from \(X\) to \(A\), then \(A\eqsim X\).
   • Let \(T=S^1\times S^1\) be the torus and let \(p\in T\). Construct a deformation retraction of \(T\setminus p\) onto a subspace homeomorphic to a wedge of two circles (i.e., a figure-eight graph).

   (Terminology note: The definition of a deformation retraction varies - Hatcher requires that there is a homotopy \(H:X\times [0,1] \to X\) such that \(H(0,x) = r(x)\) and \(H(1,x) = x\) for all \(x\) (\(H\) is a homotopy from \(r\) to \(\id_X\)) and also \(H(u,a) = a\) for all \(a\) (the points of \(A\) stay fixed throughout the homotopy). Other authors call this a strong deformation retraction.)

This problem set is for practice for the midterm. You don't need to turn it in.

1. (Hatcher, p. 38, #6) If \(s_0\in S^1\) and \(x_0\in X\), we can regard \(\pi_1(X,x_0)\) as the set of homotopy classes of maps from \(S^1\) to \(X\) that send \(s_0\) to \(x_0\). Let \([S^1,X]\) be the set of homotopy classes (sometimes called the free homotopy classes) of maps \(S^1\to X\) with no conditions on basepoints. Suppose that \(X\) is path-connected. Show that the map \(\Phi\from \pi_1(X,x_0)\to [S^1,X]\) that "forgets" basepoints is onto and that \(\Phi([f])=\Phi([g])\) if and only if \([f]\) and \([g]\) are conjugate in \(\pi_1(X,x_0)\). Hence, it induces a bijection between \([S^1,X]\) and the set of conjugacy classes of \(\pi_1(X)\).
2. A topological group is a group \(G\) equipped with a topology such that the multiplication map \(m(g,h)=gh\) and inverse map \(i(g) = g^{-1}\) are continuous. Show that if \(G\) is a topological group, then \(\pi_1(G)\) is abelian.
3. Suppose that \(X\) is simply connected. Show that for any \(p,q\in X\), any two paths from \(p\) to \(q\) are path homotopic.
4. Show that \(\pi_1(\R^3\setminus \{0\}) = 0\).