Quotients and products From the space of real numbers, form the identification space $\Bbb R/\sim$ by identifying $\frac1n\sim~n$, $\forall n$. Describe a typical neighborhood of $0$ on the space $\Bbb R/\sim$. Look at the subset $A=\bigcup_{n=1}^{\infty}(n,n+1)$. I need to show that $0\in \bar A$, but no sequence in $A$ converges to $0$.
Please help. I love math but this topology is not for me..
 A: I'll try to give some observations/hints that should direct you.  I will use the following notations:  


*

*$\mathbb{N} = \{ 1 , 2 , \ldots \}$.

*$q$ will denote the quotient mapping $\mathbb{R} \to \mathbb{R} / \mathord{\sim}$;

*for each $n \in \mathbb{N}$ I'll denote by $\hat{n}$ the $\sim$-equivalence class of $n$ (i.e., $\hat{n} = \{ \frac 1n , n \}$).


Q1: If $U \subseteq \mathbb{R} / \mathord{\sim}$ is an open neighbourhood of $0$, then $q^{-1} [ U ]$ is an open neighbourhood of $0$ in the usual topology on $\mathbb{R}$.  This means that there is an $ \epsilon > 0$ such that $( - \epsilon , + \epsilon ) \subseteq q^{-1} [ U ]$.  Note that given $n \in \mathbb{N}$ we have $\hat{n} \in U$ iff $n \in q^{-1} [ U ]$ iff $\frac 1n \in q^{-1} [ U ]$.
Q2: Completing the description to Q1 should make this quite easy.
Q3:  Suppose $\langle x_n \rangle_{n=1}^\infty$ is a sequence in $A$ converging to $0$.  Note that $\mathbb{R} / \mathord{\sim}$ is Hausdorff, and therefore $0$ is the only cluster point of the sequence.  It follows that for each $n \in \mathbb{N}$ the set $\{ n \in \mathbb{N} : | x_n - n | \leq \frac 12 \}$ is finite.  Use this to construct an open neighbourhood of $0$ containing no points of the sequence.

Added (3 March 2013)
The following image is a visualisation of the space in question.

The "jags" at the bottom of the image are supposed to represent the intervals $[ n , n + 1 ]$ (moving right-to-left).  The open neighbourhoods of the points $\hat{n} = \{ n , \frac 1n \}$ look essentially like this.

Note that it must contain a part of the interval around $\frac 1n$ and also a part of the jags from $n-1$ to $n$ and also from $n$ to $n+1$ (the latter corresponds to the preimage under the quotient mapping being an open set containing $n$).
As the numbers $\frac 1n$ get arbitrarily close to $0$, so, too, do the jags.  However the jags are also "big" in the sense that neighbourhoods of $0$ do not necessarily include any of them as a subset, as illustrated in the following image.

Of course, your standard open neighbourhood of $0$ in this space doesn't have to look as clean as this.
