Formalize definition of subbase of a topology

Question

Def.: let be $(A,B)$ a topological space, and $C \subseteq B$, "$C$ is subbasis of $B$ if $$\{X|\exists X_1,X_2,...,X_n \in C(X=\bigcap_{i=1}^n X_i)\} \text{ is basis of } B$$ Is it correct?

Answer 1

No, a basis for a topology need not be a topology at all. In fact, most often it is not. A subbasis for a topology $\tau $ on a set $S$ is simply any collection of subsets of $S$ such that the smallest topology containing all these subsets is $\tau $ itself. More concretely, $\mathcal B$ is a subbase for $\tau $ is $\mathcal B\subseteq \tau $ and every open set in $\tau $ is an arbitrary union of finite intersections of elements from $\mathcal B$.

Answer 2

No - this is almost the opposite of the spirit of the definition. You should think of $C$ as something like a generating set for $B$. All elements of $B$ are unions of intersections of elements of $C$ - but these unions and intersections will not (necessarily) be in $C$ itself, so $(A,C)$ is usually not a topological space.

For example, if $(A,B)$ is the real line with the standard topology, you could take $C$ to be the set of open intervals; this is not closed under unions, but all open sets in $\mathbb{R}$ are unions of open intervals.