Let $X = \mathbb{R}$ and $x\sim_1 y \iff x,y \in \mathbb{Z}$. Let $A_1 = X / \sim_1$.
Let $Y = [0,1]$ and $x \sim_2 y \iff x,y \in \left\{ \frac{1}{n} \; | \; n \in \mathbb{N} \right\} \cup \{ 0 \} $. Let $A_2 = Y / \sim_2$ .
I am trying to show that these two spaces are not homeomorphic and also geometrically understand why this is so. Geometrically I think of both of these two spaces as a flower with countably many leaves (empty leaves, with just the boundary) or as a Hawaiian earring. I can not see, geometrically, why these two spaces aren't the same.
The second part I have troubles is with how to formally prove they are not homeomorphic. My current idea is to show that $A_1$ is not compact, while $A_2$ is. I know that $A_2$ is compact, since it is a continuous image (by quotient projection) of a compact space $Y$, which is then itself compact. But I do not know how to show that $A_1$ is not compact.
Best Answer
One open cover of $X$ that does not have a finite subcover is given by taking the open sets $U_n \colon n \in \mathbb{Z}$ whose pre-image in $\mathbb{R}$ is the open interval $(n,n+1)$ (i.e. an open segment of each loop), Then throw in one extra open set that contains the centre of the "petal": its preimage in $\mathbb{R}$ would be something like $\bigcup_{m\in\mathbb{Z}} (m- \varepsilon, m+\varepsilon)$ for some small $\varepsilon$.
This open cover doesn't have a finite subcover because most of each loop is only in one open set so all the open sets are necessary.
The difference between the two spaces $X$ and $Y$ is what's going on at $0$ in $Y$. That point $0$ acts like a limit point of all the loops in the sense that any open set containing $0$ contains infinitely many of the loops.