I know that there are topologies that have to be defined in terms of a basis, for example, the standard topology on $\mathbb{R}$. I'm wondering if there is an examples of a topology that necessarily has to be defined in terms of a sub-basis?
Here are the relevant definitions from Munkres:
Basis:
If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis elements) such that
(1) For each $x \in X$, there is at least one basis element $B$ containing $x$.
(2) If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3 \subset B_1 \cap B_2$.
If $\mathcal{B}$ satisfies these two conditions, then we define the topology $\mathcal{T}$ generated by $\mathcal{B}$ as follows: A subset $U$ of $X$ is said to be open in $X$ (that is, to be an element of $\mathcal{T}$) if for each $x \in U$, there is a basis element $B \in \mathcal{B}$ such that $x \in B$ and $B \subset U$. Note that each basis element is itself an element of $\mathcal{T}$.
Subbasis:
A subbasis $\mathcal{S}$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $\mathcal{S}$ is defined to be the collection $\mathcal{T}$ of all unions of finite intersections of elements of $\mathcal{S}$.
Now is there an example of a topology that necessarily has to be defined in terms of a subbasis?
The product topology on an infinite Cartesian product of topological space is a candidate, but even there one can simply characterize the topology in terms of a basis. Am I right?
Best Answer
If $\mathcal{S}$ is a subbasis for $\mathcal{T}$, then the set of all finite intersections of elements of $\mathcal{S}$ is a basis for $\mathcal{T}$. So there's never going to be a context where it's much harder to describe a basis than a subbasis (although describing a subbasis might be slightly simpler).