Definition 5.5.1. Let $X$ be a topological space. A collection of subsets $\mathcal{B}$ of $X$ is called a *base for the topology on $X$* or a *basis for the topology on $X$* if the following conditions hold:

Every element $B \in \mathcal{B}$ is open in $X$.

For every open $U \subset X$ and every $x \in U$, there exists an element $B \in \mathcal{B}$ such that $x \in B \subset U$.

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