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October 16th, 2012, 02:55 AM   #1
Joined: Oct 2012

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Problem on product topology/standard topology on R^2.

Let $\mathbb{R}_{\tau}$ be the set of real numbers with topology
$\tau = \{(-x,x)| x>0\} \cup \{\emptyset, \mathbb{R}\}$
and $\mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$ be the product topology on $\mathbb{R}^2$.

a)Prove that $A = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 < 1\}$ is open in $\mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$

b)Find $\overline{A}$.Justify your answer.

c) What functions $f: {\mathbb{R}_{\tau}}^2 \rightarrow \mathbb{R}$ are continuous?
Here $\mathbb{R}$ has the standard topology and ${\mathbb{R}_{\tau}}^2 = \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}$ has the product topology.


Please help!
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vercammen is offline  
October 19th, 2012, 12:06 PM   #2
Joined: Jul 2011
From: Trieste but ever Naples in my heart! Italy, UE.

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You start to draw (U.F.O. topology or shooting topology) on so you'll ask to question a and b.
j18eos is offline  

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