In topology, a subbase (or subbasis, prebase, prebasis) for a topological space ${\displaystyle X}$ with topology ${\displaystyle \tau }$ 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.

Definition

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 two following equivalent conditions:

1. The subcollection ${\displaystyle B}$ generates the topology ${\displaystyle \tau .}$ This means that ${\displaystyle \tau }$ is the smallest topology containing ${\displaystyle B}$: any topology ${\displaystyle \tau ^{\prime }}$ on ${\displaystyle X}$ containing ${\displaystyle B}$ must also contain ${\displaystyle \tau .}$
2. The collection of open sets consisting of all finite intersections of elements of ${\displaystyle B,}$ together with the set ${\displaystyle X,}$ forms a basis for ${\displaystyle \tau .}$^[1] This means that every proper open set in ${\displaystyle \tau }$ can be written as a union of finite intersections of elements of ${\displaystyle B.}$ Explicitly, given a point ${\displaystyle x}$ in an open set ${\displaystyle U\subsetneq X,}$ there are finitely many sets ${\displaystyle S_{1},\ldots ,S_{n}}$ of ${\displaystyle B,}$ such that the intersection of these sets contains ${\displaystyle x}$ and is contained in ${\displaystyle U.}$

(If we use the nullary intersection convention, then there is no need to include ${\displaystyle X}$ in the second definition.)

For any subcollection ${\displaystyle S}$ of the power set ${\displaystyle \wp (X),}$ there is a unique topology having ${\displaystyle S}$ as a subbase. In particular, the intersection of all topologies on ${\displaystyle X}$ containing ${\displaystyle S}$ satisfies this condition. In general, however, 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 two conditions is more useful than the other.

Alternative definition

Less commonly, a slightly different definition of subbase is given which requires that the subbase ${\displaystyle {\mathcal {B}}}$ cover ${\displaystyle X.}$^[2] In this case, ${\displaystyle X}$ is the union of all sets contained in ${\displaystyle {\mathcal {B}}.}$ This means that there can be no confusion regarding the use of nullary intersections in the definition.

However, this definition is not always equivalent to the two definitions above. There exist topological spaces ${\displaystyle (X,\tau )}$ with subcollections ${\displaystyle {\mathcal {B}}\subseteq \tau }$ of the topology such that ${\displaystyle \tau }$ is the smallest topology containing ${\displaystyle {\mathcal {B}}}$, yet ${\displaystyle {\mathcal {B}}}$ does not cover ${\displaystyle X}$. (An example is given at the end of the next section.) In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies the T₁ separation axiom must be a cover of that space. But as seen below, to prove the Alexander subbase theorem,^[3] one must assume that ${\displaystyle {\mathcal {B}}}$ covers ${\displaystyle X.}$ ^{[clarification needed]}

Examples

The topology generated by any subset ${\displaystyle {\mathcal {S}}\subseteq \{\varnothing ,X\}}$ (including by the empty set ${\displaystyle {\mathcal {S}}:=\varnothing }$) is equal to the trivial topology ${\displaystyle \{\varnothing ,X\}.}$

If ${\displaystyle \tau }$ is a topology on ${\displaystyle X}$ and ${\displaystyle {\mathcal {B}}}$ is a basis for ${\displaystyle \tau }$ then the topology generated by ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle \tau .}$ Thus any basis ${\displaystyle {\mathcal {B}}}$ for a topology ${\displaystyle \tau }$ is also a subbasis for ${\displaystyle \tau .}$ If ${\displaystyle {\mathcal {S}}}$ is any subset of ${\displaystyle \tau }$ then the topology generated by ${\displaystyle {\mathcal {S}}}$ will be a subset of ${\displaystyle \tau .}$

The usual topology on the real numbers ${\displaystyle \mathbb {R} }$ has a subbase consisting of all semi-infinite open intervals either of the form ${\displaystyle (-\infty ,a)}$ or $$Zdroj:https://en.wikipedia.org?pojem=Subbase
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