Homework 03

Print preview

Additional printing options

Highlight workspace

Worksheet 3.3 Homework 03

Instructions: Complete all the exercises below and submit your work via Gradescope by Monday February 16 at 10pm.

1.

Complete one of the following tasks: either (a) read Ted Chiang’s short story Tower of Babylon and write a one-page description of the topology of the story’s world, or (b) write a one-page description of some other real or imagined object with a novel topology. Your description should, at minimum, include both the quotient and product topologies.

🔗

2.

Prove Proposition 2.2.25: If \((Z,\tau)\) and \((Z,\tau')\) are topologies on \(Z\) satisfying the universal property of Definition 2.2.24, then \(\tau=\tau'\text{.}\)

🔗

3.

In this problem, you will determine the homeomorphism type of some quotients of the plane.

🔗

(a)

For points \((x_0,y_0),(x_1,y_1)\in \mathbb R^2\text{,}\) declare

\begin{equation*} (x_0,y_0)\sim (x_1,y_1)\iff x_0+y_0^2=x_1+y_1^2. \end{equation*}

Endow \(\mathbb R^2/\sim\) with the quotient topology and determine, with proof, a familiar space with which it is homeomorphic.

🔗

(b)

Do the same thing again but with

\begin{equation*} (x_0,y_0)\sim (x_1,y_1)\iff x_0^2+y_0^2=x_1^2+y_1^2. \end{equation*}

🔗

4.

Let \(X,Y\) be spaces and give \(X\times Y\) the product topology. Prove that the projection maps \(X\leftarrow X\times Y\to Y\) are open maps.

🔗

5.

For a space \(X\) and positive integer \(n\text{,}\) we write

\begin{equation*} \operatorname{Conf}_n(X) := \{(x_1,\ldots,x_n)\in X^n\mid x_i=x_j\implies i=j\} \subseteq X^n \end{equation*}

for the configuration space of \(n\) labeled points in \(X\text{.}\) It is endowed with the subspace topology relative to \(X^n\text{.}\) The unordered configuration space of \(n\) points in \(X\) is the quotient

\begin{equation*} \operatorname{UConf}_n(X) := \operatorname{Conf}_n(X)/\sim \end{equation*}

where \(\mathbf x\sim \mathbf y\iff\) the coordinates of \(\mathbf y\) are a permutation of the coordinates of \(\mathbf x\text{.}\)

🔗

(Aside: If you know about group actions, then you can note that the permutation group on \(n\) letters, \(\mathfrak S_n\text{,}\) acts on \(\operatorname{Conf}_n(X)\) by

\begin{equation*} \sigma\cdot (x_1,\ldots,x_n) = (x_{\sigma 1},\ldots,x_{\sigma n}), \end{equation*}

and the unordered configuration space is the quotient \(\operatorname{UConf}_n(X) = \operatorname{Conf}_n(x)/\mathfrak S_n\text{.}\))

🔗