- #1

Numberphile

- 6

- 1

## Homework Statement

Let (χ,τ) be a topological space and β be a collection of subsets of χ. Then β is a basis for τ if and only if:

1. β ⊂ τ

2. for each set U in τ and point p in U there is a set V in β such that p ∈ V ⊂ U.

2. Relevant definitions

2. Relevant definitions

Let τ be a topology on a set χ and let β ⊂ τ. Then β is a basis for the topology τ if and only if every open set in τ is the union of elements of β.

## The Attempt at a Solution

I'm proving the if and only if. So for one direction, I have the following:

Suppose β ⊂ τ and for each set U in τ and point p in U, there is a set, call it V

_{p}in β such that p ∈ V

_{p}⊂ U.

Consider the union ∪ V

_{p}. Every point p in U is also in the union ∪ V

_{p}. So it follows that U ⊂ ∪ V

_{p}.

Notice every point p in the union ∪ V

_{p}is also in U. So it follows that ∪ V

_{p}⊂ U.

This shows U = ∪ V

_{p}, hence U is a union of elements of β, and therefore β is a basis.

For the other direction, this is what I attempted, but I am not sure the logic follows:

Suppose β is a basis. Then every open set U in τ is the union of elements of β, call them B

_{i}for i in some index. So U = ∪ B

_{i}, with all B

_{i}open.

I'm thinking this follows from the definition that all such collections of {B

_{i}} ⊂ τ, which implies that β ⊂ τ. (satisfying point 1)

Now Suppose there is some point p in U. Then it follows that p is in at least one such B

_{i}, call it V. Therefore there exists a set V in β such that p ∈ V ⊂ U, satisfying point 2.