Does every topology have a subbasis? I know that every topology is generated by a basis.
Is it true that every topology has a subbasis which generates it? If not, what makes it not possible to "synthesize" a subbasis out of a basis?
Having studied linear algebra, I am intuitively comfortable with the idea of basis, but it is not clear to me why we are interested in subbasis in the first place.


Thm:
Any class $\mathcal{A}$ of subsets of a non-empty set $X$ is the subbase for a unique topology on $X$. That is, finite intersections of members of $\mathcal{A}$ form a base for a topology $\mathcal{T}$ on $X$.


Proof:
We show that the class $\mathcal{B}$ of finite intersections of members of $\mathcal{A}$ satisfies the two conditions for it to be a base for a topology on $X$:
$(i)$ $X = \bigcup\{B: B \in \mathcal{B} \} $
$(ii)$ For any $G, H \in \mathcal{B}$, $\:G \cap H$ is the union of members of
$\mathcal{B}$.
Note $X \in \mathcal{B}$, since $X$ by definition is the empty intersection of members of $\mathcal{A}$; so
$X = \bigcup\{B: B \in \mathcal{B} \} $
Furthermore, if $G, H \in \mathcal{B}$, then $G$ and $H$ are finite intersections of members $\mathcal{A}$. Hence $G \cap H$ is also a finite intersection of members of $\mathcal{A}$ and therefore belongs to $\mathcal{B}$.
Accordingly, $\mathcal{B}$ is a base for a topology $\mathcal{T}$ on $X$ for which $\mathcal{A}$ is a subbase.

It is a straight forward exercise to show the uniqueness.
