My Math Forum Problem on product topology/standard topology on R^2.

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October 16th, 2012, 01:55 AM   #1
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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.

PICTURE ATTACHED!

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 October 19th, 2012, 11:06 AM #2 Member   Joined: Jul 2011 From: Trieste but ever Naples in my heart! Italy, UE. Posts: 62 Thanks: 0 You start to draw $\tau$ (U.F.O. topology or shooting topology) on $\mathbb{R}$ so you'll ask to question a and b.

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