Question

Show that B = {[a, b) ⊂ ℝ | a < b} is a basis for a topology on ℝ

Answer 1

Answer:

Remember that a set $\mathcal{B}$ is a base for some topology on $\mathbb{R}$ if satisfy the following properties:

1. $\mathbb{R}=\cup\{B: B\in\mathcal{B}\}$

2. For $B, B^* \in \mathcal{B}$, If $p\in B\cap B^*$ then exist $B_p\in\mathcal{B}$ such that $p\in B_p\subset B\cap B^*$.

Now, for $B=\{[a,b)\subset\mathbb{R}| a < b\}$ we verify the above properties:

1. It's clear that $\mathbb{R}=\cup_{a,b \text{ with }a<b}[a,b)$

2. Let $B=[a,b), B^*=[c,d) \in \mathcal{B}, p\in B\cap B^*$. Without loss of generality suppose that $a<c, b <d$. Then $c\leq p < b$, this implies that $p\in[c,b)$ and $B_p=[c,d) \in \mathcal{B}$ and $B_p\subset B\cap B^*$.

Then, B satisfy the two properties. This show that B is a basis for a topology in $\mathbb{R}$