I am confused by a lemma in my textbook. The lemma is the following:
Lemma: If $B$ is any basis for (topology) $\mathbb{T}$ then
- $T$ is the union of sets from $B$
- if $B_1,B_2\in B$ then $B_1\cap B_2$ is the union of sets from $B$
The definition of a basis for a topology in the textbook is:
Definition: A basis for a topology $\mathbb{T}$ on a set $T$ is a collection $B\subset \mathbb{T}$ such that every set in $\mathbb{T}$ is a union of sets from $B$.
The thing I don't understand is what does the second statement mean? What does "$B_1\cap B_2$ is the union of sets from $B$" exactly mean? Also, why is it called a union when it's an intersection?
Best Answer
The second part of the lemma says that the set $B_1\cap B_2$ it is also union of some subsets of $\mathcal B$.
Other lemma that you can find in some texts is the following. It can help you to understand yours.
Let $X$ a non-empty set. $\mathcal B\subset \mathcal P (X)$ is basis of any topology on $X$ if and only if:
Using the second part of this lemma you can easily prove yours.