I came across a lemma in some online notes where it says that, "Given a collection of elements of a basis ($\mathcal{B}$), they are also elements of Topology ($\mathcal{T}$) on $X$. How do we know this. I know we can define open sets in terms of basis. How can we go the other way?
I am using the following definition:
If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis elements) such that
(1)For each $x∈X$, there is at least one basis element $B$ containing $x$
(2)If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3⊆B_1∩B_2$.
If $\mathcal{B}$ satisfies these two conditions, then we define the topology $\mathcal{T}$ generated by $\mathcal{B}$ as follows: A subset $U$ of $X$ is said to be open in $X$ if for each $x∈U$, there is a basis element B∈$\mathcal{B}$ such that $x∈B$ and $B⊆U$.