Let \(\displaystyle \mathbb{R}_{\tau}\) be the set of real numbers with topology
\(\displaystyle \tau = \{(-x,x)| x>0\} \cup \{\emptyset, \mathbb{R}\}\)
and \(\displaystyle \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}\) be the product topology on \(\displaystyle \mathbb{R}^2\).
a)Prove that \(\displaystyle A = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 < 1\}\) is open in \(\displaystyle \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}\)
b)Find \(\displaystyle \overline{A}\). Justify your answer.
c) What functions \(\displaystyle f: {\mathbb{R}_{\tau}}^2 \rightarrow \mathbb{R}\) are continuous?
Here \(\displaystyle \mathbb{R}\) has the standard topology and \(\displaystyle {\mathbb{R}_{\tau}}^2 = \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}\) has the product topology.
PICTURE ATTACHED!
Please help!
\(\displaystyle \tau = \{(-x,x)| x>0\} \cup \{\emptyset, \mathbb{R}\}\)
and \(\displaystyle \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}\) be the product topology on \(\displaystyle \mathbb{R}^2\).
a)Prove that \(\displaystyle A = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 < 1\}\) is open in \(\displaystyle \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}\)
b)Find \(\displaystyle \overline{A}\). Justify your answer.
c) What functions \(\displaystyle f: {\mathbb{R}_{\tau}}^2 \rightarrow \mathbb{R}\) are continuous?
Here \(\displaystyle \mathbb{R}\) has the standard topology and \(\displaystyle {\mathbb{R}_{\tau}}^2 = \mathbb{R}_{\tau} \times \mathbb{R}_{\tau}\) has the product topology.
PICTURE ATTACHED!
Please help!
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