Topology problems

**chath** · October 16th 2012, 07:16 AM

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).

**Plato** · October 16th 2012, 07:32 AM

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 $\mathbb{Q}$ is dense in $\mathbb{R}$

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

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

Finish it off.

**chath** · October 16th 2012, 07:55 AM

thank you very much friend...

**chath** · October 16th 2012, 07:58 AM

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

**Plato** · October 16th 2012, 08:16 AM

Originally Posted by chath

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

Let $\mathcal{B}(q;r)=\{(x,y): d[(x,y),(c,d)]<r.$

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

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

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