Define τ⊂ P(ℝ) as follows:

τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .

prove that τ is a topology on ℝ .

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- May 25th 2010, 11:01 PMblblprove that τ is the topology on ℝ
Define τ⊂ P(ℝ) as follows:

τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .

prove that τ is a topology on ℝ . - May 25th 2010, 11:29 PMDrexel28
Is this $\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:(-\pi,\pi)\subseteq U\right\}$? What's wrong? clearly $\displaystyle \varnothing,\mathbb{R}\in T$ if $\displaystyle (-\pi,\pi)\subseteq U_\alpha$ then $\displaystyle (-\pi,\pi)\subseteq\bigcup_{\alpha\in\mathcal{A}}U_\ alpha$ and similarly for the intersection. Or, is this $\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:\ {-\pi,\pi\}\subseteq U\right\}$? It's the same.