I'm reading Munkres 2nd ed. and I'm a bit confused by the language. When you consider a topology $\mathscr{T}$, are the following statements equivalent?

The topology $\mathscr{T}$ is generated by the collection of sets $\mathscr{B}$

The topology $\mathscr{T}$ has as its basis the collection of sets $\mathscr{B}$


Saying that $\mathscr B$ is a basis is a stronger claim than saying that it generates the topology.

When you say that $\mathscr B$ is a basis for the topology, you're claiming that every open set is a union of sets from $\mathscr B$.

When you say that $\mathscr B$ generates the topology, you may need to take intersections too in order to produce all open sets. (And, as an extreme case, $\bigcup\mathscr B$ may not be the entire space, but the entire space is still open, as it is required to).


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