Subbase

In topology, a subbase (or subbasis, prebase, prebasis) for the topology τ of a topological space (X, τ) is a subcollection ${\displaystyle B}$ of ${\displaystyle \tau }$ that generates ${\displaystyle \tau ,}$ in the sense that ${\displaystyle \tau }$ is the smallest topology containing ${\displaystyle B}$ as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

Subbase is a weaker notion than that of a base for a topology.

Let ${\displaystyle X}$ be a topological space with topology ${\displaystyle \tau .}$ A subbase of ${\displaystyle \tau }$ is usually defined as a subcollection ${\displaystyle B}$ of ${\displaystyle \tau }$ satisfying one of the three following equivalent conditions:

If we additionally assume that ${\displaystyle B}$ covers ${\displaystyle X}$, or if we use the nullary intersection convention, then there is no need to include ${\displaystyle X}$ in the third definition.

If ${\displaystyle B}$ is a subbase of ${\displaystyle \tau }$, we say that ${\displaystyle B}$ generates the topology ${\displaystyle \tau .}$ This terminology originates from the explicit construction of ${\displaystyle \tau }$ from ${\displaystyle B}$ using the second or third definition above.

Elements of subbase are called subbasic (open) sets. A cover composed of subbasic sets is called a subbasic (open) cover.

For any subcollection ${\displaystyle S}$ of the power set ${\displaystyle \wp (X),}$ there is a unique topology having ${\displaystyle S}$ as a subbase; it is the intersection of all topologies on ${\displaystyle X}$ containing ${\displaystyle S}$. In general, however, the converse is not true, i.e. there is no unique subbasis for a given topology.

Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set ${\displaystyle \wp (X)}$ and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the three conditions is more useful than the others.