Difference between basis and subbasis in a topology?

I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?

Bases and subbases "generate" a topology in different ways. Every open set is a union of basis elements. Every open set is a union of finite intersections of subbasis elements.

For this reason, we can take a smaller set as our subbasis, and that sometimes makes proving things about the topology easier. We get to use a smaller set for our proof, but we pay for it; with a subbasis we need to worry about finite intersections, whereas we did not have to worry about that in the case of a basis.

Consider $S=\{\{0,1\},\{0,2\}\}$. What is the topological space $T(S)$ generated by $S?$ By definition, $S$ will then be a subbasis of $T(S)$.

Well, we want to all requirements to hold true and find that $T(S) = \{\emptyset, \{0\}, \{0,1\}, \{0,2\}, \{0,1,2\}\}$ (check this!).

Is $S$ a basis? No, because you cannot write $\{0\}$ as a union of any elements in $S$.

So you see that subbasis and basis are two different notions, even for a very basic example.

A subbasis can be thought of, and is actually defined to be, the "smallest set that becomes my topological space if I complete it under the property of being a topological space, i.e. fulfiling the axioms of topological space".

The two terms are related nevertheless. Every basis is a subbasis, and in one of the equivalent definitions of subbasis you will find that you already get a basis from your subbasis.

The collection of sets $(-\infty,b)$ and $(a,\infty)$ for $a,b\in \mathbb{R}$ constitute a sub-basis for the standard topology on $\mathbb{R}$. The collection of sets $(a,b)$ for $a,b\in \mathbb{R}$ constitute a basis for the standard topology on $\mathbb{R}$. I suggest that you look at the definitions of "basis" and "sub-basis" and convince yourself that the claims that I have made are correct; this is probably the best way to answer your question.

The basic idea is that a basis is the collection of all finite intersections of sub-basis elements. The open sets in a topology are all possible unions of basis elements. So, the open sets in a topology are all possible unions of finite intersections of sub-basis elements.

I hope that answers your question!