Q1: Let $(X,\mathcal{T})$ be a topological space. What is correct for a base $\mathcal{B}$ of a topology $\mathcal{T}$?

A1: $\mathcal{B} \in \mathcal{T}$

A2: $\mathcal{B} \in X$

A3: $\mathcal{B} \subseteq X$

A4: $\mathcal{B} \subseteq \mathcal{T}$

Q2: Let $(X,\mathcal{T})$ be a topological space. What is always correct for every base $\mathcal{B}$ of a topology $\mathcal{T}$?

A1: Every open set can be written as an intersection of the sets from $\mathcal{B}$.

A2: Every open set can be written as a union of the sets from $\mathcal{B}$.

A3: Every open set can be written as a finite union of the sets from $\mathcal{B}$.

Q3: Does every topological space have a basis?

A1: Yes!

A2: No!

A3: Only if it is the discrete topological space.