Describe a typical neighborhood of $\mathbb{Z}$ in the decomposition space.

Question

In the space $\mathbb{R}$ of real numbers, identify the integers to a point. This is the space $\mathbb{R}/\mathbb{Z}$. Describe a typical neighborhood of $\mathbb{Z}$ in the decomposition space.

I need help.

Answer 1

Let $q:\Bbb R\to\Bbb R/\Bbb Z$ be the quotient map, and let $p$ be the point of $\Bbb R/\Bbb Z$ corresponding to $\Bbb Z$; i.e., $q^{-1}\big[\{p\}\big]=\Bbb Z$. Suppose that $U$ is an open nbhd of $p$. Then by definition $\Bbb Z\subseteq q^{-1}[U]$, and $q^{-1}[U]$ is open in $\Bbb R$. Thus, you’re looking for open nbhds $U$ of $\Bbb Z$ in $\Bbb R$, and you know what they look like.

It helps to have a mental picture of $\Bbb R/\Bbb Z$; this question and my answer to it may help. I’ll leave it to you to come up with a more precise description, but informally an open nbhd of $p$ has to include an open interval around $p$ on every one of the circles.

(By the way, this space is quite a bit simpler to visualize than the one in your other question here.)