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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!

  • 0
    got stuck here.2012-10-16
  • 2
    You got stuck *before starting*?2012-10-16
  • 0
    need to know what to start with2012-10-16
  • 2
    looking up the *definitions* and trying to apply them..2012-10-16

1 Answers 1