# Thread: prove that τ is the topology on ℝ

1. ## 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 .

2. 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 $\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}-\pi,\pi)\subseteq U\right\}$? 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.