# Topology problems

• Oct 16th 2012, 07:16 AM
chath
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).
• Oct 16th 2012, 07:32 AM
Plato
Re: Topology problems
Quote:

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.
• Oct 16th 2012, 07:55 AM
chath
Re: Topology problems
thank you very much friend...:)
• Oct 16th 2012, 07:58 AM
chath
Re: Topology problems
can you please briefly explain me the thing that you've done...
• Oct 16th 2012, 08:16 AM
Plato
Re: Topology problems
Quote:

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)]

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

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