What is the difference between these two notions of basis for a topology?

Question

From Introduction to Topological Manifolds by John M. Lee:

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If we have collection of subsets of some set $X$, why do we need to check that it satisfies the two conditions in the proposition? Is it not enough to check that it satisfies the definition of basis for the topology of $X$?

I see that in the definition we start with a topological space $X$, whereas in the proposition we just start from the set $X$, but I cannot tell exactly what the difference is.

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For this exercise:

do I want to show that it satisfies the definition above, or that it satisfies the proposition?

Answer 1

There are two situations that must be discerned:

1) $X$ is some topological space so a topology $\tau_X\subseteq\wp(X)$ is fixed. Then we can have bases for this specific topology. These are collections $\mathcal B$ that satisfy the conditions (i) and (ii) that are mentioned in the definition above.

2) There is a set $X$ but not yet a fixed topology on it. Then we can wonder: what are the characteristics of a collection $\mathcal B$ that can serve as a basis of some topology? These characteristics are mentioned in (i) and (ii) in the proposition below.

If it is known already that $\mathcal B$ indeed serves as a basis of some topology then you do not have to check anymore whether the collection satisfies the conditions mentioned in the proposition. That will automatically be the case.

Answer 2

You need condition (i) to prove that $X$ is an open set.

You need condition (ii) to prove that the intersection of two basis elements is an open set.