1. ## Topology problems

can somebody help me with these two problems immediately???

1.Let D_p be any open disc in R×R with centre p=(a,b) and radius δ>0.Show that there exist an open disc (Dbar) such that
(i)the centre of (Dbar) has rational coordinates.
(ii) the radius of (Dbar0 is rational.
(iii)p is an element of (Dbar) and (Dbar) is a subdet of D_p

2.Let f:R→R be a continuous function such that f(q)=sin⁡q for q∈Q.Find the value of f(π/4).

2. ## Re: Topology problems

Originally Posted by chath
can somebody help me with these two problems immediately???
1.Let D_p be any open disc in R×R with centre p=(a,b) and radius δ>0.Show that there exist an open disc (Dbar) such that
(i)the centre of (Dbar) has rational coordinates.
(ii) the radius of (Dbar0 is rational.
(iii)p is an element of (Dbar) and (Dbar) is a subdet of D_p
This is a simple problem.
Use the fact that $\displaystyle \mathbb{Q}$ is dense in $\displaystyle \mathbb{R}$

$\displaystyle \left( {\exists r \in \mathbb{Q}} \right)\left[ {0 < r < \frac{\delta }{2}} \right]$

$\displaystyle \left( {\exists q = (c,d) \in \mathbb{Q} \times \mathbb{Q}} \right)\left[ {D(p,q) < r} \right]$

Finish it off.

3. ## Re: Topology problems

thank you very much friend...

4. ## Re: Topology problems

can you please briefly explain me the thing that you've done...

5. ## Re: Topology problems

Originally Posted by chath
can you please briefly explain me the thing that you've done...
Let $\displaystyle \mathcal{B}(q;r)=\{(x,y): d[(x,y),(c,d)]<r.$

$\displaystyle \mathcal{B}(q;r)$ is a ball with rational coordinates at its centre and contains the point $\displaystyle p$.

Then $\displaystyle p\in\mathcal{B}(q;r)~\&~\overline{\mathcal{B}(q;r) }\subset D.$