Definition of a subbasis of a topology

[V0ODO0CH1LD]

One of the definitions of a subbasis $\mathcal{S}$ of a set $X$ is that it covers $X$. Then the collection of all unions of finite intersections of elements of $\mathcal{S}$ make up a topology $\mathcal{T}$ on $X$. That means the collection of all finite intersections of elements of $\mathcal{S}$ is a basis $\mathcal{B}$ for the topology $\mathcal{T}$.

But one of the defining characteristics of a basis is that it also must cover $X$, although if the subbasis is the collection of all singletons in $X$, which definitely covers $X$, then the basis $\mathcal{B}$ would have only the empty set; wouldn't it?

 

[economicsnerd]

If \mathcal S = \{ \{x\}: \enspace x\in X\} as you describe, then the set of all finite intersections of members of \mathcal S is just \mathcal B = \mathcal S \cup \{\emptyset, X\}. This is a basis for the topology \mathcal T = \left\{ \bigcup \hat{\mathcal B}: \enspace \hat{\mathcal B} \subseteq \mathcal B \right\}= \{A: \enspace A\subseteq X \} which some call the discrete topology.

 

[V0ODO0CH1LD]

How does the intersection of singletons of a set give you the whole set? I can see that they give you the empty set, but not the whole set itself. Or are you just throwing the whole set into complete the basis??

 

[R136a1]

How does the intersection of singletons of a set give you the whole set? I can see that they give you the empty set, but not the whole set itself. Or are you just throwing the whole set into complete the basis??

It doesn't give $X$. But the process to form a topology given a subbasis $\mathcal{S}$ is the following:
1) First adjoin $\emptyset$ and $X$.
2) Take all finite intersections
3) Take all unions

So this is why he had the set $X$, since you need to adjoin it according to (1).

However, you seem to have a bit of another definition of a subbasis. You demand that a subbasis covers $X$. This is not the standard definition, I believe. But if you follow your definition than the steps are:
1) Adjoin $\emptyset$
2) Take all finite intersections
3) Take all unions.

 

[economicsnerd]

It's not that important, but I was using the convention that the intersection of no sets is the whole space. i.e. Given a universe X, for any collection \mathcal A \subseteq 2^X of sets, one common definition of the intersection is \bigcap \mathcal A:= \{x\in X: \enspace x\in A \text{ for every } A\in\mathcal A\}. If this is the definition you like, then \bigcap\emptyset=X. Other people adopt the convention that "\bigcap \emptyset" is just undefined.