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:

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

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

There are also:

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