prove that τ is the topology on ℝ

• May 25th 2010, 11:01 PM
blbl
prove 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 PM
Drexel28
Quote:

Originally Posted by blbl
Define τ⊂ P(ℝ) as follows:
τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .
prove that τ is the topology on ℝ .

Is this $T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:(-\pi,\pi)\subseteq U\right\}$? What's wrong? clearly $\varnothing,\mathbb{R}\in T$ if $(-\pi,\pi)\subseteq U_\alpha$ then $(-\pi,\pi)\subseteq\bigcup_{\alpha\in\mathcal{A}}U_\ alpha$ and similarly for the intersection. Or, is this $T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:\ {-\pi,\pi\}\subseteq U\right\}$? It's the same.