A *base* for a [[Topology|topological space]] is a way of *generating* its topology from something simpler. # Definitions > [!NOTE] Definition > A *base* (or *basis*) for a topology $\mathcal{T}$ on $X$ is a subset $\mathcal{B} \subset \mathcal{T}$ such that every set in $\mathcal{T}$ is the union of some sets from $\mathcal{B}$, i.e. for all $U \in \mathcal{T}$, there exists $\mathcal{C}_U \subset \mathcal{B}$ such that $U=\bigcup_{B \in C_U} B$. The following is an equivalent characterisation of base $\mathcal{B}$ for $\mathcal{T}$. See [[Recognition of Topological Bases]]. > [!NOTE] Definition > A subset $\mathcal{B}$ of $\mathcal{T}$ is a basis for $\mathcal{T}$ iff the following conditions are met: > (B1) $\mathcal{B}$ is an (open) [[Set covers|cover]] of $X$. > (B2) For any $B_{1}, B_{2} \in \mathcal{B}$ then $B_{1} \cap B_{2}$ is the union of some sets from $\mathcal{B}$. # Examples **Example**: Singleton sets form a base for the discrete topology. **Example**: $\{ (n, n+2) \mid n\in \mathbb{Z} \}$ is an open cover for $\mathbb{R}$ but is clearly not a basis for any topology on $\mathbb{R}$.