---
tag: 0CHR35YNJQ
kind: definition
name: Subbasis
msc: [54A05]
defines: 09S8DV34Y0
uses: [0CNQXVBHCX, 06EX0FHBTF, 0839XNQXDK]
sources:
  - url: https://en.wikipedia.org/wiki/Subbase
    locator: Definition
---

## Definition

Let $X$ be a set and let $\mathcal{S}$ be a set of subsets of $X$.
The topology generated by $\mathcal{S}$ is the smallest [[topology|0CNQXVBHCX]] on $X$ that contains $\mathcal{S}$. It is the [[intersection|0839XNQXDK]] of all topologies on $X$ that contain $\mathcal{S}$.
If $(X, \tau)$ is a topological space and $\tau$ is the topology generated by $\mathcal{S}$, then $\mathcal{S}$ is a subbasis of $\tau$.

## Remarks

The open sets of the generated topology are the unions of finite intersections of members of $\mathcal{S}$. Here the empty intersection is $X$.
Thus the finite intersections of members of $\mathcal{S}$ form a [[basis|06EX0FHBTF]] of the generated topology.
The open rays $(-\infty, a)$ and $(b, \infty)$, with $a, b \in \mathbb{R}$, form a subbasis of the usual topology of $\mathbb{R}$.
The sets $\pi_i^{-1}(U)$, with $U$ open, form a subbasis of the [[product topology|04S9HX3GAP]].

## Explanations

### L1 human

A subbasis is a set of open sets that you want to have. The topology is the smallest one that makes them all open.
You get all open sets by taking finite intersections of them and then arbitrary unions.