Due Wednesday, November 20, in class

Section 54 problems 3*, 5, 6, 7
Section 55 problems 1, 3

Homework 11 Solutions

Due November 13, Wednesday, in class

*ed problems are recommended only

Section 51 problem 3
Section 52 problems 3, 4, 6*
Section 53 problems (1, 2, 3)*, 4, 5, 6
edit: (on 6(a), just prove that two of the given properties lift)

Due Wednesday, November 6, in class

*ed problems are recommended

Section 32 problems 3, 4, 5*, 6, 7*
Section 33 problems 2, 7, 8

Due Wednesday, October 30, in class

*ed problems are recommended and not graded

Section 30 problems 2*, 3, 5, 7*, 8*, 11, 12*, 13*
Section 31 problems 1*, 2, 3*, 4*, 5, 6

Due October 23 in class

The *ed problems are recommended.

Section 26 problems 6, 11
Section 27 problems 4, 6*
Section 28 problems 3, 6*, 7*
Section 29 problems 6, 7*, 8*

Due Wednesday, October 9

* denoted recommended (not graded) problems

A. Let \(X\) be the subset of \(\mathbb{R}^2\): \(X = \{[0,1] \times 0\} \cup \{1/n \times [0,1]: n\in\mathbb{Z}_+\}\cup \{0\times[0,1]\}\). Show that \(X\) is connected and path-connected, but neither locally connected nor locally path-connected.

Section 23 problems 1*, 2*, 3*, 8 and 9
Section 24 problems 1*, 2, 3, and 8*
Section 25 problems 3*, 4*, 7*, and 8

Due October 2 in class

Section 20 problems 8a, 10
Section 21 problems 4, 6
Section 22 problems 2, 3, 4

then

A. Let \(t\) denote the taxicab metric on \(\mathbb{R}^n\), given by \(t((x_1,\ldots, x_n),(y_1,\ldots,y_n)) = \sum_{i=1}^n \left|x_i – y_i\right| \). Show that \(t\) and the euclidean metric induce the same topology on \(\mathbb{R}^n\).

B. Define \(\sim\) on \(\mathbb{R}\) by \(x\sim y\) whenever \(x-y \in\mathbb{Z}\). The quotient \(\mathbb{R}/\sim\)is also denoted as \(\mathbb{R}/\mathbb{Z}\). Does \(\mathbb{R}/\mathbb{Z}\) look like any topological space that we are familiar with? Justify your answer. Secondly, define a partition \(\mathbb{R}^\ast\) consisting of the singletons \(\{x\}\) if \(x\not\in \mathbb{Z}\) and the set \(\mathbb{Z}\). Do you think the quotient space \(\mathbb{R}^\ast\) is homeomorphic to \(\mathbb{R}/\mathbb{Z}\) (Do your best to answer this last question…it will be difficult to prove this rigorously. Appeal to your intuition here)?

C. Recall from class the line with two origins. It inherits a quotient topology from \(\mathbb{R}\times 1 \sqcup \mathbb{R}\times -1 \subset \mathbb{R}^2\), and it inherits the subspace topology from \(\mathbb{R}^2\). Compare and contrast the two topologies on the line with two origins.

D. Is a quotient of a Hausdorff space necessarily Hausdorff? Prove it or provide a counterexample.

Due September 25 in class

Section 18 problems 2, 5, 8, 9
Section 19 problems 3, 6, 7
Section 20 problems 3

* denotes a suggested problem. You should know how to complete a * problem, but it will not be graded (so please do not hand it in).

Due Wednesday, September 18

A. Consider \(\mathbb{R} \times \mathbb{R}\) with the lexicographical (dictionary) order topology. (a) What topology does \(\mathbb{R}\times 0\) inherit as a subspace? (b) What topology does \(0\times \mathbb{R}\) inherit as a subspace?

Section 16 problem 6

Section 17 problems 3, 4, 6, 8*, 9, 11*, 15*

Due September 11

Section 13, problems 1, 4 (solution), 5, 7, 8