Base of a topology
Let $X$ be a non-empty set. A collection $\mathscr{B}$ of subsets of $X$ is a base for some topology on $X$ if it satisfies two conditions:
- $\mathscr{B}$ covers $X$. That is, every point of $X$ belongs to at least one member of $\mathscr{B}$.
- If $B_1,B_2\in\mathscr{B}$ and $x\in B_1\cap B_2$, then there is a $B_3\in\mathscr{B}$ such that $x\in B_3\subseteq B_1\cap B_2$.
These two conditions are exactly what is needed to ensure that
$$\mathscr{T}=\Big\{\bigcup\mathscr{A}:\mathscr{A}\subseteq\mathscr{B}\Big\}$$
is a topology on $X$. In words, the collection of all unions of members of $\mathscr{B}$ is a topology on $X$, the topology generated by the base.
Note that a topology may have many different bases. The topology $\big\{\varnothing,\{a\},\{b\},\{a,b\}\big\}$ on the set $\{a,b\}$ has the following bases:
- $\big\{\varnothing,\{a\},\{b\},\{a,b\}\big\}$
- $\big\{\{a\},\{b\},\{a,b\}\big\}$
- $\big\{\varnothing,\{a\},\{b\}\big\}$
- $\big\{\{a\},\{b\}\big\}$
Conditions (1) and (2) above are the easiest way to characterize the families of sets that are bases for some topology on $X$. If you already have a topology $\mathscr{T}$ on $X$, you can say simply that a subset $\mathscr{B}$ of $\mathscr{T}$ is a base for $\mathscr{T}$ if and only if every member of $\mathscr{T}$ (i.e., every open set in the space $\langle X,\mathscr{T}\rangle$) is a union of members of $\mathscr{B}$.
Yes, $\big\{\varnothing,\{a\},\{c\},X\big\}$ is a base for a topology on $X$: it satisfies the two conditions given at the beginning of this answer. The topology that it generates is
$$\big\{\varnothing,\{a\},\{c\},\{a,c\},\{a,b,c\}\big\}\;.$$