Let \(X\) be a set and \(\mathcal{B}\) a family of subsets. The family \(\mathcal{B}\) is a basis for some topology on \(X\) if and only it satisfies the following conditions:

1. \(\bigcup_{B\in \mathcal{B}} B=X\)
2. If \(B_1, B_2\in \mathcal{B}\) and \(x\in B_1\cap B_2\), then there is \(B_3\in \mathcal{B}\) such that \(x\in B_3\subseteq B_1\cap B_2\).

The generated topology is unique and is called topology generated by \(\mathcal{B}\). [1, Proposition 2.44]

References Link to heading

1. J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7