Basis for a topology lemma

general-topology

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:

  1. $X=\cup_{B\in \mathcal{B}}B$
  2. $\forall B_1, B_2\in\mathcal B, \ \forall x\in B_1\cap B_2 \ \exists B_3 \mid x\in B_3\subset B_1\cap B_2$.

Using the second part of this lemma you can easily prove yours.

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