Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

We have two spaces $X=\{(x,1/n):n\neq 0, n\in\mathbb{Z}, x\in\mathbb{R}\}$ and $Y=\{(x,n):n\neq 0, n\in\mathbb{Z}, x\in\mathbb{R}\}$. On both spaces we introduce the equivalent relation $(x,y)\sim (x',y')$ if $x=x'$ and $y=y'$ or $x=x'=0$. That is, all points on the $y$ axis are collapsed to the same point.

We are asked whether $X/\sim$ and $Y/\sim$ are homeomorphic in quotient topologies.

It is easy to show that the original spaces are homeomorphic. However, I don't know how to answer the question about the quotient spaces.

My guess is that they might not be homeomorphic and some problem might occur at the $origin$ but I am not sure.

Any hint would be helpful! Thanks!

share|cite|improve this question
up vote 4 down vote accepted

Once you've shown that $X$ and $Y$ are homeomorphic, it's almost immediate to show that $X / \sim $ and $ Y / \sim $ are homeomorphic :

if $f$ is the natural homeomorphism between $X$ and $Y$ just notice that $x \sim y $ if and only if $f(x) \sim f(y)$ . Then $g$ the induced fonction between $X / \sim $ and $ Y / \sim $ is continuous(by definition of the quotient topology) and one to one. Do the same with $f^{-1}$ and then $g$ is a homeomorphism.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.