Basis of a topology

Definition

Let $(X, \tau)$ be a topological space. A basis of $\tau$ is a set $\mathcal{B} \subseteq \tau$ such that each $U \in \tau$ is a union of members of $\mathcal{B}$. Equivalently, $\mathcal{B} \subseteq \tau$ and for each $U \in \tau$ and each $x \in U$ there is $B \in \mathcal{B}$ with $x \in B \subseteq U$.

Remarks

A family $\mathcal{B}$ of subsets of a set $X$ is a basis of some topology on $X$ if and only if two conditions hold. First, the members of $\mathcal{B}$ cover $X$. Second, for all $B_1, B_2 \in \mathcal{B}$ and $x \in B_1 \cap B_2$ there is $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$. This topology is then unique. The open intervals form a basis of the usual topology of $\mathbb{R}$. The intervals with rational endpoints form a countable basis. In a metric space, the open balls form a basis of the metric topology.

Explanations

L1 human

A basis is a set of building blocks: each open set is a union of blocks. In the plane, you can build each open set from open discs, or from open squares.