I've looked on this site but no answer seems to clarify my doubt or maybe I need someone to dumb it down a little. What I understand is that a basis is the collection of open sets whose union generates the topology while the subbasis is the collection of sets whose union of finite intersections generates the topology. Now I don't know how much to believe that because according to me the basis need not contain the entire set on which the topology is described while I did find such a description somewhere. I then decided to illustrate it for myself to try and understand but here too I got stuck:
Let $X$ be a set on which a topology $\mathcal{T}$ is defined
$\mathcal{T} = \{ \varnothing, \{1,2\}, \{2,3\}, \{2\}, \{1,2,3\}, \{1,2,3,4\} \}$.
In this case is
$\mathcal{B} = \{ \{1,2\}, \{2,3\}, \{2\}, \{1,2,3\}, \{4\} \}$?
Because if so $X$ doesn't belong to it which is what I read however according to this Wikipedia quote it should
The collection of open sets consisting of all finite intersections of elements of subbasis $\mathcal{S}$ together with the set $X$, forms a basis for $\mathcal{T}$.
But I don't see any reason why $X$ should belong to it.
Further what would the subbasis look like? According to the above definition I should have some element in $\mathcal{S}$ whose intersection with another element gives the element $\{1,2,3\}$ in $\mathcal{B}$ but I see not two elements that can produce this intersection. Please help.
Best Answer
B is a base of topology $\;T\;$ on a space $\;X\;$ = any open set in $\;X\;$ ( i.e., any element of $\;T\;$) can be expressed as a union of element in $\;B\;$ .
S is a sub base of topology $\;T\;$ on a space $\;X\;$ = any open set in $\;T\;$ can be a expressed as a union of finite intersections of elements in $\;S\;$ .
Thus for example, $\;B:=\{\;(a,b)\;/\;a,b\in\Bbb R\;\}\;$ is a base for the usual, Euclidean topology of the real line $\;\Bbb R\;$ , and $\;S:=\{\;(-\infty,a)\,,\,(b,\infty)\;/\;a,b\in\Bbb R\;\}\;$ is a sub base for the same topology.
Yet this is one concept that can vary from author to author...perhaps because it is not that useful (and thus it is not that important) in general...