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October 16th, 2012, 01:55 AM  #1 
Newbie Joined: Oct 2012 Posts: 4 Thanks: 0  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! Please help! 
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 (U.F.O. topology or shooting topology) on so you'll ask to question a and b.


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problem, product, topology, topology or standard 
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