Homework Problems

Homework will be due once a week. This page will be updated every time a new problem set is posted.

Your answers should always be written in narrative form; you should of course include relevant computations and equations, but these should be situated in the framework of some overall narrative that explains your reasoning, and they should always be naturally integrated into the structure of your writing. You should follow standard rules of composition: write in complete sentences, include appropriate punctuation, group sentences with a common theme into a paragraph, provide appropriate narrative transition when your reasoning takes a significant turn, etc. When writing formal proofs, you should not use abbreviations like $\forall$, $\exists$, $\Rightarrow$. It is acceptable to use the notation $\in$ when describing elements in a formal proof (e.g., "Since $2 \in \mathbb{Z}$, we see that ..."). As a general rule, a mathematical symbol should never start a sentence or follow punctuation; the exception is that you can write a mathematical symbol after a colon which is announcing the beginning of a list of mathematical objects. Of course your work should be legible and neat. The overall guiding principle in your writing is to remember that your work is meant to be read by a a skeptical peer; your job is to write a (logically) convincing argument to this audience.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

The two parts of this assignment will be submitted in via separate Google forms that show up with each assignment. You work for Part A should be collected into one PDF, and your work for Part B should be collected in another PDF. Please be sure you've written your solutions in a way that allow you to submit your solutions for Part A separate from your solutions for Part B!

Assignment 1 (due Friday, October 30)

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Part A

1. Let $(X,\mathcal{T})$ be a topological space, and let $A \subseteq X$. Define $\widehat{\mathcal{T}} = \{U \cap A: U \in \mathcal{T}\}$. Prove that $\widehat{\mathcal{T}}$ is a topology on $A$.
2. Let $X$ be given. Suppose that $\mathcal{I}$ is some index set, and for each $i \in \mathcal{I}$ we have some topology $\mathcal{T}_i$ on $X$.
   1. Show that $\bigcap_{i \in \mathcal{I}} \mathcal{T}_i$ is a topology on $X$.
   2. Show that there exists a unique topology $\mathcal{T}$ contained in all $\mathcal{T}_i$ which has the property that if $\mathcal{T_0}$ is a topology with $\mathcal{T}_0 \subseteq \mathcal{T}_i$ for all $i \in \mathcal{I}$, then $\mathcal{T}_0 \subseteq \mathcal{T}$.

      [N.B. The fancy way to view this result is to say that there exists a ``largest" topology contained in all the $\mathcal{T}_i$; this topology plays a role that's analogous to the greatest lower bound of a set of real number.]

Part B

3. Suppose that $\mathcal{B}$ is a basis, and $\mathcal{T}_{\mathcal{B}}$ is the topology generated by $\mathcal{B}$. Prove that $$\mathcal{T}_\mathcal{B} = \mathop{\bigcap_{\mathcal{B} \subseteq \mathcal{T}}}_{\mathcal{T} \text{ is a topology}}\mathcal{T}.$$
4. Recall that we wrote $\mathcal{T}_{302}$ for the "usual" topology for $\mathbb{R}$ in analysis. Let $$\mathcal{B} = \{(a,b): a,b \in \mathbb{Q} \text{ and }a < b\}$$ $$\mathcal{C} = \{[a,b):a,b \in \mathbb{Q} \text{ and }a < b\}$$ $$\mathcal{D} = \{[a,b): a,b \in \mathbb{R} \text{ and }a < b\}.$$
   1. Show that $\mathcal{T}_\mathcal{B} = \mathcal{T}_{302}$. [Feel free to cite familiar properties of $\mathbb{R}$ that you learned in 302...but be sure to be explicit in those citations!]
   2. Show that $\mathcal{T}_\mathcal{D}$ is strictly finer than $\mathcal{T}_{\mathcal{C}}$.
   3. Show that $\mathcal{T}_\mathcal{D}$ is strictly finer than $\mathcal{T}_\mathcal{B}$.

Solutions are posted here.

Assignment 2 (due Friday, November 6)

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Part A

1. Two metrics $d_1$ and $d_2$ on a set $X$ are called equivalent if there exists some $\alpha,\beta>0$ so that for all $x,y \in X$ we have $$\alpha d_1(x,y) \leq d_2(x,y) \leq \beta d_1(x,y).$$ Prove that $\mathcal{T}_{d_1} = \mathcal{T}_{d_2}$. [Hint: You know bases for both of these topologies.]
2. Define the function $d:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}_{\geq 0}$ by $$d(\mathbf{x},\mathbf{y}) = \sum_{i=1}^n |x_i-y_i|.$$ This is called the taxicab metric on $\mathbb{R}^n$.
   1. Show that the taxicab metric is a metric.
   2. Draw $B_d((1,1),2)$. (Include a brief justification)
   3. Show that the topology induced by the taxicab metric is the same as the usual topology on $\mathbb{R}^n$.
3. Suppose that $Y \subseteq X$. Let $\mathcal{T}$ be a topology on $X$, and let $\widehat{\mathcal{T}}$ be the topology on $Y$ given by $$\widehat{\mathcal{T}} = \{Y \cap U: U \in \mathcal{T}\}.$$ (You saw in homework 1 that $\widehat{\mathcal{T}}$ is a topology on $Y$; it is called the subspace topology.)
   1. Show that if $D$ is open in $Y$ (meaning: $D \in \widehat{\mathcal{T}}$) and $Y$ is open in $X$ (meaning: $Y \in \mathcal{T}$), then $D$ is open in $X$.
   2. Give an example of a spaces $Y$ and $X$, a topology $\mathcal{T}$ on $X$, and a subset $D \subseteq Y$ so that $D$ is open in $Y$, but $D$ is not open in $X$.

Part B

4. Let $(X,\mathcal{T})$ be a topological space. The closed sets relative to $\mathcal{T}$ are the sets whose complement is open.
   1. Show that the arbitrary intersection of closed sets is closed, and that a finite union of closed sets is closed.
   2. Give an example of an infinite union of closed sets that is closed.
   3. Give an example of an infinite union of closed sets that is not closed.
5. Let $(X,\mathcal{T})$ be a topological space. The closure of a set $A \subseteq X$, denoted $\bar A$, and the interior of the set $A$, denoted $A^\circ$ are $$\bar A = \mathop{\bigcap_{A \subseteq K}}_{K \text{ is closed}} K \qquad \qquad \text{ and } \qquad \qquad A^\circ = \mathop{\bigcup_{U \subseteq A}}_{U \text{ is open}} U.$$ The boundary of $A$, denoted $\text{Bd}(A)$, is $$\text{Bd}(A) = \bar A \cap \overline{X-A}.$$
   1. Show that $A^\circ \cap \text{Bd}(A) = \emptyset$.
   2. Show that $\bar A = A^\circ \cup \text{Bd}(A)$
   3. Show that $\text{Bd}(A) = \emptyset$ iff $A$ is both open and closed.
   4. Show that $U$ is open iff $\text{Bd}(U)=\bar U-U$.

Solutions are posted here.

Assignment 3 (due Friday, November 13 Saturday, November 14)

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Part A

1. In this problem, $(X_1,\mathcal{T}_1)$ and $(X_2,\mathcal{T}_2)$ are two topological spaces, and $X_1 \times X_2$ is endowed with the product topology $\mathcal{T}$.
   1. Prove that it is not necessarily the case that $\mathcal{T} = \{U_1 \times U_2: U_1 \in \mathcal{T}_1,U_2 \in \mathcal{T}_2\}$ by giving a specific example of spaces $(X_1,\mathcal{T}_1)$ and $(X_2,\mathcal{T}_2)$ and a subset $A \in \mathcal{T}$ so that $A \neq U_1 \times U_2$ for any $U_1 \in \mathcal{T}_1$ and $U_2 \in \mathcal{T}_2$.
   2. Let $\pi_1:X_1 \times X_2 \to X_1$ be defined by $\pi_1(x_1 \times x_2) = x_1$. Show that if $U \subseteq X_1 \times X_2$ is open, then $\pi_1(U)$ is open (in $X_1$).

      [Note: here we use the notation $f(U)$ to mean $\{f(u): u \in U\}$.]

2. Let $(X,\mathcal{T})$ be a topological space. Let $\mathcal{I}$ be an index set, and for each $i \in \mathcal{I}$ let $A_i$ be a subset of $X$.
   1. Show that $\displaystyle \bigcup_{i \in \mathcal{I}} \bar A_i \subseteq \overline{\bigcup_{i \in \mathcal{I}} A_i}$
   2. Show that if $|\mathcal{I}| < \infty$, then $\displaystyle \overline{\bigcup_{i \in \mathcal{I}} A_i} \subseteq \bigcup_{i \in \mathcal{I}} \bar A_i$.
   3. Give an example where $\displaystyle \overline{\bigcup_{i \in \mathcal{I}} A_i} \not\subseteq \bigcup_{i \in \mathcal{I}} \bar A_i$.
   [Hint: in the two sets you're considering, one is definitely closed, but the other might not be.]
3. Suppose that $(X_1,\mathcal{T}_1)$ and $(X_2,\mathcal{T}_2)$ are Hausdorff spaces. Prove that $X_1 \times X_2$ is also a Hausdorff space.

   [NB. Here we haven't specified that topology on $X_1 \times X_2$, but given the context the reasonabe interpretation is that $X_1 \times X_2$ has the product topology. Moving forward, when we don't say otherwise, product spaces will be given the product topology.]

Part B

4. Consider the following bases in $\mathbb{R}$: $$\mathcal{B}_1 = \{(a,b): a < b, a,b \in \mathbb{R}\} $$ $$\mathcal{B}_2 = \{[a,b): a < b, a,b \in \mathbb{R}\} $$ $$\mathcal{B}_3 = \{(a,b]: a < b, a,b \in \mathbb{R}\} $$ $$\mathcal{B}_4 = \{(a,b): a < b, a,b \in \mathbb{R}\} \cup \{(a,b)-K: a < b, a,b \in \mathbb{R}\},$$ where in the last basis the set $K$ is defined as $K = \{\frac{1}{n}: n \in \mathbb{N}\}$. For each $1 \leq i \leq 4$, let $\mathcal{T}_i$ be the topology generated by $\mathcal{B}_i$.

   Determine (with proof) whether the sequence $(\frac{1}{1},\frac{1}{2},\frac{1}{3},\cdots)$ has a limit in each of these 4 topologies. (Note: each of these topologies is Hausdorff, so you don't have to worry about checking for multiple limits.)

   [NB. Here, when we say "limit" we mean the topological definition of limit.]

5. Suppose that $X$ has two topologies: $\mathcal{T}_1$ and $\mathcal{T}_2$. To help distinguish $(X,\mathcal{T}_1)$ and $(X,\mathcal{T}_2)$, we will write $X_1$ for the first and $X_2$ in place of the second. (This way we can quickly "see" which topology we're considering on the set $X$.) Let $\text{id}:X_1 \to X_2$ be the function defined by $\text{id}(x) = x$.
   1. Show that $\text{id}$ is continuous if and only if $\mathcal{T}_1$ is finer than $\mathcal{T}_2$.
   2. Show that $\text{id}$ is a homeomorphism if and only if $\mathcal{T}_1 = \mathcal{T}_2$.
6. Suppose that $f:X_1 \to X_2$ is a continuous function, and let $A \subseteq X$.
   1. Critique the following "proof" that if $x \in A'$, then $f(x) \in f(A)'$:

      Suppose that $x \in A'$. This means that $x \in \overline{A-\{x\}}$, and so for any $U \in \mathcal{N}_x$ we have some $u \neq x$ with $u \in U \cap A$. To show that $f(x) \in f(A)'$, we need to show that $f(x) \in \overline{f(A)-\{f(x)\}}$, which means for every $V \in \mathcal{N}_{f(x)}$ we need to find some $\alpha \neq f(x)$ with $\alpha \in f(A) \cap V$. So let $V \in \mathcal{N}_{f(x)}$ be given. Since $f$ is continuous, we know that $f^{-1}(V)$ is open in $X_1$, and so $f^{-1}(V) \in \mathcal{N}_x$ by definition of neighborhood. But then from the property for $x$ listed above, we know there exists some $a_V \neq x$ so that $a_V \in A \cap f^{-1}(V)$. But this means $f(a_V) \in f(A) \cap V$. Choosing $\alpha = f(a_V)$ gives the desired result.

   2. Give a specific example of topological spaces $X_1$ and $X_2$, a subsets $A \subseteq X_1$, a cluster point $x \in A'$, and a continuous function $f:X_1 \to X_2$ so that $f(x)\not\in f(A)'$.

      [Hint: if you can find where the argument above breaks down, it could give you a hint of how to construct an example like this.

Solutions are posted here.

Assignment 4 (due Saturday, November 21)

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Part A

1. Suppose that $X$ is a set with a metric $d$. As usual, the corresponding topology is denoted $\mathcal{T}_d$. Suppose that $X$ also has another topology $\mathcal{T}$. To distinguish the spaces defined by these two topologies, we'll write $X_1$ when we consider $X$ with the topology $\mathcal{T}_d$, and $X_2$ when we consider $X$ with the topology $\mathcal{T}$.
   1. Show that $d:X_1 \times X_1 \to \mathbb{R}$ is continuous.
   2. Suppose that $d:X_2 \times X_2 \to \mathbb{R}$ is continuous. Prove that $\mathcal{T} \supseteq \mathcal{T}_d$.
2. Suppose that $X$ and $Y$ are metric spaces, with metrics denoted $d_X$ and $d_Y$ respectively. A function $f:X \to Y$ is called an isometry if for all $x_1,x_2 \in X$ we have $$d_X(x_1,x_2) = d_Y(f(x_1),f(x_2)).$$

   Show that if $f:X \to Y$ is an isometry, then $X$ is homeomorphic to the image $f(X)$.

3. Suppose that $X$ is a metric space with metric $d$. If $(x_n)$ is a sequence of points and $x \in X$, then we say that $x$ is a limit of $(x_n)$ under the metric $d$ if for any $\varepsilon>0$ there exists some $N \in \mathbb{N}$ so that for all $n \geq N$ we have $d(x_n,x)<\varepsilon$. Show $x$ is a limit of $(x_n)$ according to the topological definition if and only if $x$ is a limit of $(x_n)$ under the metric $d$.

Part B

4. Suppose that $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ are functions, and define $f_1 \times f_2: X_1 \times X_2 \to Y_1 \times Y_2$ by $$(f_1\times f_2)(x_1 \times x_2) = f_1(x_1) \times f_2(x_2).$$ Prove that if $f_1$ and $f_2$ are continuous, then $f_1 \times f_2$ is continuous.
5. Let $X$ be a topological space. Suppose that $\mathcal{I}$ is an index set, and for each $i \in \mathcal{I}$ we have some set $A_i$ within $X$ satisfying the condition that $\bigcup_{i \in \mathcal{I}} A_i = X$. For a function $f:X \to Y$, we denote the function $f|_{A_i}$ by $f_i$ for simplicity.
   1. Suppose that $|\mathcal{I}|<\infty$, that each $f_i$ is continuous, and that each $A_i$ is closed. Prove that $f$ is continuous.
   2. Give an example where $|\mathcal{I}| = |\mathbb{N}|$, where each $f_i$ is continuous, each $A_i$ is closed, but $f$ is not continuous.
6. Let $P:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be defined by $P(x \times y) = x+y$. Prove that $P$ is continuous.

Solutions are posted here.

Assignment 5 (due Saturday, December 5)

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Part A

1. The purpose of this problem is to argue that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic.
   1. Suppose that $X_1$ and $X_2$ are topological spaces which are homeomorphic, and let $f:X_1 \to X_2$ be a homeomorphism. Prove that for any $x_1 \in X_1$, the spaces $X_1 -\{x_1\}$ and $X_2-\{f(x_1)\}$ are also homeomorphic.
   2. Let $x \in \mathbb{R}$ and $\mathbf{z} \in \mathbb{R}^2$ be given. Use a topological property that we've already discussed to argue that $\mathbb{R}-\{x\}$ and $\mathbb{R}^2-\{\mathbf{z}\}$ are not homeomorphic.
   3. Prove that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic.
2. Suppose that $k:X \to Y$ is continuous.
   1. Show that if $f_0$ and $f_1$ are paths in $X$ that are path homotopic, then $k \circ f_0$ and $k \circ f_1$ are also path homotopic.
   2. Suppose that $f$ and $g$ are paths in $X$ with $f(1) = g(0)$. Prove that $k \circ (f*g) = (k \circ f)* (k \circ g)$. [Here, the operation $*$ is and product operation on paths from Day 24.]
3. ($\star$) Suppose that $X$ has the discrete topology. Prove that all paths are constant functions.

Part B

In class we studied path homotopies, and I mentioned they are a special case of a more general idea called homotopy. If $f$ and $\hat f$ are continuous functions from $X$ to $Y$, then a homotopy from $f$ to $\hat f$ is a continuous function $F:X \times I \to Y$ so that $$F(x,0) = f(x) \text{ and }F(x,1) = \hat f(x) \text{ for all }x \in X.$$ If a homotopy from $f$ to $\hat f$ exists, then we say that $f$ is homotopic to $\hat f$, and we write $f \simeq \hat f$. It is a fact (proven in the text, but that you don't have to reproduce) that "homotopic to" is an equivalence relation on the set of continuous functions from $X$ to $Y$. For a continuous function $f:X \to Y$, let $[f]$ denote the equivalence class of $f$ under the equivalence relation "homotopic to." That is, $$[f] = \{\hat f:X \to Y: \hat f \text{ is continuous and }f \simeq \hat f\}.$$

In these problems, we'll study some further properties of homotopies.

4. Give an example of a space $X$ and two paths $f$ and $g$ on $X$ so that $f$ and $g$ are homotopic, but not path homotopic.
5. Suppose that $f$ and $\hat f$ are continuous functions from $X$ to $Y$, and that $g$ and $\hat g$ are continuous functions from $Y$ to $Z$. Prove that if $f \simeq \hat f$ and $g \simeq \hat g$, then $g \circ f \simeq \hat g \circ \hat f$.
6. 1. UPDATED! Suppose $0 < a < b < 1$, and that $f$ is a path in $X$. Define $g:I \to X$ by $g(s) = f(a+s(b-a))$. Prove that $f$ is homotopic to $g$.
   2. Let $X$ be any topological space that is path connected (i.e., for any two points $x_0,x_1 \in X$, there is some path $f:I \to X$ so that $f(0)=x_0$ and $f(1)=x_1$). Prove that any two paths in $X$ are homotopic. [Hint: Use the fact that $X$ is path connected to create a path from the terminal point of the first path to the initial point of the second path; you can string these three paths together to make one giant path, then use the previous part of this problem.]

Solutions are posted here.

Assignment 6 (due Friday, December 11)

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Part A

1. A subset $A \subseteq \mathbb{R}^n$ is called star convex if for some $a_0 \in A$, all the line segments joining $a_0$ to other points of $A$ lie in $A$.
   1. ($\star$) Give an example of a set that is star convex, but is not convex. (Recall that $A$ is called convex if the segment connecting any two points of $A$ lies in $A$.)
   2. Prove that if $A$ is star convex and $a_0 \in A$ is any point, then $\pi_1(A,a_0)$ is trivial.
2. Suppose that $\alpha$ is a path in $X$ that starts at $x_0$ and ends at $x_1$, and let $\beta$ be a path in $X$ that starts at $x_1$ and ends at $x_2$. For the path $\gamma = \alpha * \beta$, show that $\hat \gamma = \hat \beta \circ \hat \alpha$.
3. Let $h:X \to Y$ be a continuous function. For a point $x \in X$, we write $(h_x)*$ for the homomorphism induced by $h$ relative to the base point $x$. That is, $(h_x)_*:\pi_1(X,x) \to \pi_1(Y,h(x))$ is defined by $(h_x)_*([f]) = [h \circ f]$. (It's a fact from Day 28 (that you don't need to prove here) that $h_{x_0}$ is a well-defined homomorphism of groups.)

   Let $x_0,x_1 \in X$ be given. Suppose that $\alpha$ is some path in $X$ from $x_0$ to $x_1$, and let $\beta = h \circ \alpha$. Prove that $\hat \beta \circ (h_{x_0})_* = (h_{x_1})_* \circ \hat \alpha$.

Part B

There is no Part B!

Solutions are posted here.