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Fudgin [204]
3 years ago
8

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

Mathematics
1 answer:
loris [4]3 years ago
6 0

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

2. Let B=[a,b), B^*=[c,d) \in \mathcal{B}, p\in B\cap B^*. Without loss of generality suppose that a. 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}

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