Homework 04

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Worksheet 3.4 Homework 04

Instructions: Complete all the exercises below and submit your work via Gradescope by Monday February 23 at 10pm.

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1.

Let \(f,g\colon X\to Y\) be two continuous functions.

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(a)

Prove that

\begin{align*} f\times g\colon X\times X \amp \longrightarrow Y\times Y\\ (x,x') \amp \longmapsto (f(x),g(x')) \end{align*}

is continuous.

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(b)

Prove that the diagonal map

\begin{align*} \Delta\colon Y \amp \longrightarrow Y\times Y \\ y \amp \longmapsto (y,y) \end{align*}

is continuous.

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(c)

Suppose that \(Y\) separates points. Prove that

\begin{equation*} \{x\in X\mid f(x)=g(x)\} \end{equation*}

is closed.

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(d)

Prove that \(X\) separates points if and only if

\begin{equation*} \{(x,x)\mid x\in X\}\subseteq X\times X \end{equation*}

is closed (with respect to the product topology on \(X\times X\)).

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2.

Prove that the line with two origins \(\mathbb R\amalg_{\mathbb R\smallsetminus \{0\}}\mathbb R\) does not separate points. You might proceed in the following fashion: let \(0_+,0_-\) denote the two origins. Let \(U_{\pm}\) denote open neighborhoods of \(0_{\pm}\text{,}\) respectively. Construct the canonical continuous function \(\mathbb R\amalg_{\mathbb R\smallsetminus \{0\}}\mathbb R\to \mathbb R\) and use it to show that \(U_+\cap U_-\) is nonempty.

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3.

Let \(i\colon S^1\hookrightarrow D^2\) denote the continuous inclusion of \(S^1\) as the boundary of \(D^2\text{.}\) Let \(P\) denote the pushout of

\begin{equation*} S^1\times D^2\xleftarrow{\operatorname{id}_{S^1}\times i} S^1\times S^1 \xrightarrow{i\times \operatorname{id}_{S^1}} D^2\times S^1. \end{equation*}

Construct a continuous bijection \(P\to S^3\text{.}\)

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It will likely prove helpful to view \(S^3\) as a subspace of \(\mathbb C^2\cong \mathbb R^4\) and think about functions like

\begin{align*} S^1\times D^2 \amp \longrightarrow S^3 \\ (w,z) \amp \longmapsto \frac{(w,z)}{\sqrt{1+|z|^2}}. \end{align*}

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4.

Recall that a partially ordered set (or poset) \((P,\le)\) is a set \(P\) equipped with a reflexive, antisymmetric, transitive binary relation \(\le\text{.}\) Given a poset \((P,\le)\text{,}\) let \(\tau_{\le}\) denote the collection of subsets of \(P\) that are upwards closed, that is, \(U\in \tau_{\le}\) :iff \(x\in U\) and \(x\le y\in P\) implies \(y\in U\text{.}\)

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