Paper 4, Section II, H

(1) Prove that if $X$ is a compact topological space, then a closed subset $Y$ of $X$ endowed with the subspace topology is compact.

(2) Consider the following equivalence relation on $\mathbb{R}^{2}$ :

$\left(x_{1}, y_{1}\right) \sim\left(x_{2}, y_{2}\right) \Longleftrightarrow\left(x_{1}-x_{2}, y_{1}-y_{2}\right) \in \mathbb{Z}^{2}$

Let $X=\mathbb{R}^{2} / \sim$ be the quotient space endowed with the quotient topology, and let $p: \mathbb{R}^{2} \rightarrow X$ be the canonical surjection mapping each element to its equivalence class. Let $Z=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sqrt{2} x\right\} .$

(i) Show that $X$ is compact.

(ii) Assuming that $p(Z)$ is dense in $X$, show that $\left.p\right|_{Z}: Z \rightarrow p(Z)$ is bijective but not homeomorphic.

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