# HW1

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Homework Assignment 1

Required reading. Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read all of Munkres chapter 2.

Solve the following problems. (But submit only the underlined ones). In Munkres' book (Topology, 2nd edition), problems 4, 8 on pages 83-84, problems 4, 8 on page 92, problems 6, 7, 13 on page 101, problems 9, 11, 12, 13 on page 112, problems 6, 7 on page 118 and problems 3, 8 on pages 126-128. Also solve (but don't submit) the following

Problem. Let C be the "Cantor set", the closure of the set of real numbers in [0,1] whose expansion to base 3 doesn't contain the digit 1 (e.g., $\frac34=0.2020202020202\ldots$ is in, but $\frac12=0.11111111\ldots$ is out). Prove that C (taken with the topology induced from $\mathbb R$) is homeomorphic to $\{0,1\}^{\mathbb N}$ (taken with the product topology).

Due date. This assignment is due at the end of our class on Tuesday, October 11, 2005.

Above the line, edit only if you are sure.

Feel free to edit below the line.

## Problems for submission

### $\S$13, Problem 8, p. 83

1. Apply Lemma 13.2 to show that the countable collection $\mathcal{B}=\{(a,b)\;|\;a < b, a,b\in\mathbb Q\}$ is a basis that generates the standard topology on $\mathbb R$.
2. Show that the collection $\mathcal{C}=\{[a,b)\;|\; a < b, a,b\in\mathbb Q\}$ is a basis that generates a topology different from the lower limit topology on $\mathbb R$.

### $\S$16, Problem 4, p. 92

A map $f \colon X \to Y$ is said to be an open map if for every open set U of X, the set f(U) is open in Y. Show that $\pi_1 \colon X \times Y \to X$ and $\pi_2 \colon X \times Y \to Y$ are open maps.

### $\S$17, Problem 13, p. 101

Show that X is Hausdorff iff the diagonal $\Delta=\{x \times x \;|\; x \in X\}$ is closed in $X \times X$.

### $\S$18, Problem 13, p. 112

Let $A \subset X$; let $f \colon A \to Y$ be continuous; let Y be Hausdorff. Show that if f may be extended to a continuous function $g \colon \bar A \to Y$, then g is uniquely determined by f.

### $\S$19, Problem 6, p. 118

Let $\mathbf{x_1}, \mathbf{x_2}, \ldots$ be a sequence of the points of the product space $\prod X_\alpha$. Show that this sequence converges to the point $\mathbf{x}$ iff the sequence $\pi_\alpha(\mathbf{x_1}), \pi_\alpha(\mathbf{x_2}), \ldots$ converges to $\pi_\alpha(\mathbf{x)}$ for each α. Is this fact true if one uses the box topology instead of the product topology?

### $\S$20, Problem 3, p. 126

Let X be a metric space with metric d.

1. Show that $d \colon X \times X \to \mathbb R$ is continuous.
2. Let X' denote a space having the same underlying set as X. Show that if $d \colon X' \times X' \to \mathbb R$ is continuous, then the topology of X' is finer than the topology of X.

One can summarize the result of this exercise as follows: If X has a metric d, then the topology induced by d is the coarsest topology relative to which the function d is continuous.

## Other problems

### $\S$13, Problem 4, p. 83

1. If $\{\mathcal{T}_\alpha\}$ is a family of topologies on X, show that $\bigcap\mathcal{T}_\alpha$ is a topology on X. Is $\bigcup\mathcal{T}_\alpha$ a topology on X?
2. Let $\{\mathcal{T}_\alpha\}$ be a family of topologies on X. Show that there is a unique smllest topology on X containing all the collections $\mathcal{T}_\alpha$, and a unique largest topology contained in all of $\mathcal{T}_\alpha$.
3. If X = {a,b,c}, let $\mathcal{T}_1=\{\emptyset, X, \{a\}, \{a, b\}\}$ and $\mathcal{T}_2=\{\emptyset, X, \{a\}, \{b, c\}\}$. Find the smallest topology containing $\mathcal{T}_1$ and $\mathcal{T}_2$. Find the largest topology contained in $\mathcal{T}_1$ and $\mathcal{T}_2$.

### $\S$16, Problem 8, p. 92

If L is a straight line in the plane, describe the topology L inherits as a subspace of $\mathbb R_l \times \mathbb R$ and as a subspace of $\mathbb R_l \times \mathbb R_l$. In each case it is a familiar topology.

### $\S$17, Problem 6, p. 101

Let A, B and Aα denote subsets of a space X. Prove the following:

1. If $A \subset B$, then $\bar A \subset \bar B$.
2. $\overline{A \bigcup B} = \bar A \bigcup \bar B$.
3. $\overline{\bigcup A_\alpha} \supset \bigcup \bar A_\alpha$; give an example where equality fails.

### $\S$17, Problem 7, p. 101

Criticize the following "proof" that $\overline{\bigcup A_\alpha} \subset \bigcup \overline{A}_\alpha$: if Aα is a collection of sets in X and if $x \in \overline{\bigcup A_\alpha}$, then every neighbourhood U of x intersects $\bigcup A_\alpha$. Thus U must intersect some Aα, so that x must belong to the closure of some Aα. Therefore, $x \in \bigcup \overline{A}_\alpha$.

When trying to show a set was closed, many answers used the equivalent property ${\overline A = A}$ , and even ${\overline {X-A} = X-A}$ to show a set is open. For this assignment at least, these properties were far easier to prove either directly or by using that the pre-image of an open or closed set is open or closed respectively. Also ${\overline {\{x:d(x,x_0)<\epsilon\}}\ne\{x:d(x,x_0)\le\epsilon\}}$, even in subset topologies of Euclidean space.