# Problems on topological and metric spaces.

I need help on the following problems:

Q1: Consider the map $f:(\Bbb R\times\Bbb R,\tau)\to(\Bbb R,\gamma)$ given by $f(x\times y)=|x-y|$,where $\tau$ is the product topology and $\gamma$ is the usual topology. Find $f(1),f(0),f^{-1}(0)$ and $f^{-1}[0,1)$.

For this problem, I don’t know how to use the given definition to find say $f(1)$, also I find it difficult to figure out $f^{-1}$.

Another one i need help on: I read somewhere that the set of rational numbers $\Bbb Q$ as a subspace of $\Bbb R$ (with usual topology) does not have discrete topology since the point $0$ is not an open set in $\Bbb Q$. ¨

Why is this argument true? I thought that $0$ was rational. Thanks.

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f(1) doesn't seem to make sense to me. (Is R the set of real numbers?) The topology on Q is given by 'open balls' (= open intervals) on Q. Can you find a radius r and a midpoint q such that the open ball of radius r about the point q (i.e. the interval (q-r, q+r)) contains 0 and no other points? – Billy May 11 '13 at 20:05
@Billy:yes R is the set of real number,Thanks – user77362 May 11 '13 at 20:19

The number $1$ is not (at least not without further conventions) an element of $\mathbb R\times \mathbb R$, hence I understand why you have problems finding out what $f(1)$ should be.
For $f^{-1}$, you need to find all $(x,y)$ such that $|x-y|$ ...
For the totally unrelated about $\mathbb Q$, note that $\frac1n\to 0$.