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.

1. Let \((X,d)\) be a metric space, let \(x\in X\), let \(r>0\). Show that \(B(x,r)=\{y\in X\mid d(x,y) < r\}\) is an open set.
2. Recall the usual \(\epsilon\)-\(\delta\) definitions of limits and continuity.
   1. Show that the \(\epsilon\)-\(\delta\) definition of limits is equivalent to the open set definition.
   2. Show that the \(\epsilon\)-\(\delta\) definition of continuity is equivalent to the open set definition.
3. 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 \(f(x_n) \to f(a)\).
4. Let \(X\) be a topological space. Suppose that \(a,b\in X\), \(a\ne b\), but that every sequence such that \(x_n \to a\) also satisfies \(x_n \to b\). What does this imply about \(X\), \(a\), and \(b\)?

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. (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 \(\R\).

For the next two problems, we say that a set \(S\subset \R\) is cofinite if there are only finitely many real numbers that are not in \(S\). Let \(\mathscr{T}\) be the set of cofinite subsets of \(\R\) along with the empty set.

3. • Show that \(\mathscr{T}\) is a topology.
   • Show that the sequence \(1,2,3,\dots\) converges to any real number in the cofinite topology.
4. Let \(\mathbf{A}=(a_i)_{i=1}^\infty\) be a sequence of real numbers. For a real number \(x\), let the multiplicity \(m_x(\mathbf{A})\) be the number of \(i\)'s such that \(a_i=x\). Find a necessary and sufficient condition based on multiplicities that characterizes the sequences such that \(a_i\to 0\) in the cofinite topology.

1. (Munkres, p. 100, #3) Show that if \(A\) is closed in \(X\) and \(B\) is closed in \(Y\), then \(A\times B\) is closed in \(X\times Y\).
2. 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.

3. Recall that the lower-limit topology on \(\R\) is the topology generated by the half-open intervals \(\{[a,b)\mid a < b\}\). We denote \(\R\) with the lower-limit topology by \(\R_{\ell}\).

   Let \(P_{\ell} = \R_{\ell} \times \R_{\ell}\). Let \(X\) be a straight line in \(P_\ell\). Describe the subspace topology that \(X\) inherits from \(P_\ell\). How does this depend on the direction of \(X\)?

4. Let \((X,d)\) be a metric space. Equip \(X\) with the metric space topology and \(X\times X\) with the product topology. Show that \(d\from X\times X \to \R\) is continuous.
5. Suppose that \(x,y \in X\) have the property that if \(U\) is a neighborhood of \(x\) and \(V\) is a neighborhood of \(y\), then \(U\cap V \ne \emptyset\). Let \(f\from X\to \R\) be a continuous map. Show that \(f(x)=f(y)\).

1. Let \(A\subset X\). Show that \(\partial A = \partial(X\setminus A)\). Show that \(A\setminus \partial A\) is an open set and that \(A\cup \partial A\) is a closed set.
2. 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.
3. (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.
4. (Munkres, p. 158, #9) Assume that \(\R\) is uncountable. Show that if \(A\) is a countable subset of \(\R^2\), then \(\R^2\setminus A\) is path-connected. (Hint: How many lines are there passing through a given point of \(\R^2\)?)

1. Let \(X\) be a Hausdorff topological space. Show that finite unions of compact subsets of \(X\) are compact. Show that arbitrary intersections of compact subsets of \(X\) are compact.
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. 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 if and only if \(X\) is totally bounded and every Cauchy sequence in \(X\) converges.

4. A closed map is a function \(f\from X\to Y\) such that \(f(K)\) is closed whenever \(K\subset X\) is closed. Suppose that \(X\) and \(Y\) are topological spaces and let \(p\from X\times Y\to X\) be the map \(p(x,y)=x\). Show that if \(Y\) is compact, then \(p\) is closed. Find an example of a non-compact \(Y\) such that \(p\) is not closed.

1. Show that quotients of compact spaces are compact.
2. Show that a finite-dimensional CW complex is connected if and only if its 1-skeleton is connected.
3. • Construct a CW complex homeomorphic to \(\R\).
   • Let \(X\) be the union of \(S^2\) and the diameter connecting the north pole to the south pole. Construct a CW complex homeomorphic to \(X\).

4. In class we defined change of basepoint maps for the fundamental group; if \(\gamma\) is a path from \(x\) to \(y\), then \(b_\gamma\from \pi_1(X,y)\to \pi_1(X,x)\) is the map \(b_\gamma([\alpha])=[\gamma\cdot \alpha\cdot \bar{\gamma}]\).

   Suppose that \(\gamma\) and \(\gamma'\) are both paths from \(x\) to \(y\). Show that \(b_\gamma\) and \(b_{\gamma'}\) differ by a conjugation. That is, there is an \(h\in \pi_1(X,x)\) such that for all \(g\in \pi_1(X,y)\), \[b_\gamma(g)=h b_{\gamma'}(g)h^{-1}.\]

5. (Hatcher, p. 38, #6) We can regard \(\pi_1(X,x_0)\) as the set of basepoint-preserving homotopy classes of maps \((S^1,s_0)\to (X,x_0)\). 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 \(\pi_1(X)\).

1. Hatcher, p. 38, #8
2. Hatcher, p. 39, #10
3. Let \(\gamma\from [0,1]\to S^1\) be a loop based at \(x_0\in S^1\) and let \(\omega\from \pi_1(S^1,x_0)\to \Z\) be the isomorphism we constructed in class. Suppose that \(y\in S^1\) and that \(\#\gamma^{-1}(y)=n\), where \(\#\gamma^{-1}(y)\) is the number of points in the preimage of \(y\). Show that \(|w(\gamma)|\le n\).
4. Recall that we showed that \(\R\) is not homeomorphic to \(\R^2\) by comparing \(\R\setminus\{0\}\) and \(\R^2\setminus \{0\}\). Show that \(\R^2\) is not homeomorphic to \(\R^3\).

I'll post part 2 of the problem set on Thursday; for now, do as many of these problems as you can, and we'll work on some more examples with van Kampen's theorem this week.

1. • Show that the complement of finitely many points in \(\R^3\) is simply-connected (has trivial fundamental group).
   • Show that the complement of a $k$–dimensional subspace of \(\R^n\) is simply-connected if \(k\le n-3\).
2. Hatcher, p. 52, #2
3. Hatcher, p. 53, #4
4. Hatcher, p. 53, #10

5. Given spaces \(X\) and \(Y\), a subset \(Z\subset Y\), and a map \(f\from Z\to X\), let \(X\cup_f Y=(X\sqcup Y)/\sim\), where \(z\sim f(z)\) for all \(z\in Z\). We say that this is \(X\) with \(Y\) attached along the map \(f\).

   Let \(X\) be a space and let \(D\) be the unit $n$–ball for some \(n\ge 3\). Let \(\gamma:\partial D\to X\). Show that \(\pi_1(X\cup_\gamma D)\cong \pi_1(X)\).

6. In this problem, we show that the intersection conditions in van Kampen's Theorem are necessary.
   • Find a space \(X\) and two path-connected open subsets \(U\), \(V\) that share a basepoint \(x_0\) such that \(X=U\cup V\), but the canonical map \(\Phi\from \pi_1(U,x_0)\ast \pi_1(V,x_0)\to \pi_1(X,x_0)\) is not surjective.
   • Find a space \(X\) and three path-connected open subsets \(A_1\), \(A_2\), and \(A_3\) that share a basepoint \(x_0\) such that \(X=A_1\cup A_2\cup A_3\) and \(A_i\cap A_j\) is path-connected for all \(i\) and \(j\), but the kernel of the canonical map \(\Phi\from \pi_1(A_1,x_0)\ast \pi_1(A_2,x_0)\ast \pi_1(A_3,x_0) \to \pi_1(X,x_0)\) is not normally generated by elements of the form \([\gamma]_i[\gamma]_j^{-1}\). (It may help to consider examples where \(A_i\cap A_j\) is simply-connected for all \(i\ne j\).)

1. Hatcher, page 53, #11
2. The real projective plane \(\R P^2\) is the quotient of the sphere \(S^2\) by the equivalence relation \(x\sim -x\). It can be given the structure of a CW complex with 4 vertices, 6 edges, and 3 faces by identifying opposite vertices, edges, and faces in the surface of a cube. Use this structure to show that \(\pi_1(\R P^2)=\Z/2\Z\).
3. Let \(X\) be connected and let \(p\from \tilde{X}\to X\) be a covering space. Suppose that \(p^{-1}(x)\) is finite for every \(x\in X\). Show that the number of points in \(p^{-1}(x)\) is the same for every \(x\in X\). This is called the number of sheets of the cover.
4. Let \(f\from X\to X\) be a homeomorphism and let \(T_f\) be the mapping torus constructed in the problem from Hatcher above. Show that \(X\times \R\) is a covering space of \(T_f\) by finding a covering map \(p\from X\times \R \to T_f\). What is the image \(p_*(\pi_1(X\times\R))\subset \pi_1(T_f)\)?

1. Graphs:
   • Show that the fundamental group of a connected graph with \(n\) vertices and \(m\) edges is the free group with \(m-n+1\) generators (a free group with rank \(m-n+1\)).
   • Let \(X\) be the wedge of \(k\) circles, with fundamental group \(F_k\), and let \(p\from \tilde{X}\to X\) be a connected cover of \(X\) with \(n\) sheets. Let \(H=p_*(\pi_1(\tilde{X}))\subset F_k\). Find a formula relating the index of \(H\) in \(F_k\) to its rank as a free group.
2. Hatcher, p. 79, #4
3. Hatcher, p. 79, #6
4. (A lifting criterion for unbased covering spaces) Suppose that \(p\from \tilde{X}\to X\) is a path-connected covering space and that \(Y\) is a path-connected and locally path-connected space. If \(f\from (Y,y_0)\to (X,x_0)\), show that \(f\) lifts to a map \(\tilde{f}\from Y\to \tilde{X}\) if and only if a conjugate of \(f_*(\pi_1(Y))\) is a subgroup of \(p_*(\pi_1(\tilde{X}))\).