0
$\begingroup$

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 $\mathscr{B}$

$\endgroup$
1
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.