Let X = R. Define an equivalence relation ~ on X by x ~ y ifand only if x – y is an integer. Prove that the quotient space X* ishomeomorphic to the circle S'.-

Answered: Let X = R. Define an equivalence…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let X = R. Define an equivalence relation ~ on X by x ~ y if
and only if x – y is an integer. Prove that the quotient space X* is
homeomorphic to the circle S'.
-

Expert Solution

Step 1

Let $(X, 𝛕_{X})$ be a topological space, and let $~$ be an equivalence relation on $X$ .

The quotient set, $X^{*} = \frac{X}{~}$ is the set of equivalence classes of elements of X. The equivalence class of $x \in X$ is denoted by $[x]$ .

The quotient space under $~$ is the quotient set $X^{*}$ equipped with the quotient topology.

i.e., the topology whose open sets are the subsets $U \subset X^{*}$ such that $\{x \in X : [x] \in U\}$ is open.

That is, $𝛕_{X^{*}} = {U \subset X^{*} : \{x \in X : [x] \in U\}}$

The space $X^{*}$ is the quotient space of $X$ .

Let X and Y be a topological spaces. Let $f : X \to Y$ be a bijection. If both the function $f$ and the inverse function $f^{- 1} : Y \to X$ are continuous, then $f$ is a homeomorphism .

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