# Definition: Subbasis and Basis of Topology

Let $(X,\mathcal O)$ be a topological space. * A basis of $(X,\mathcal O)$ is a subset $B\subseteq\mathcal O$ such that every element $O\in\mathcal O$ equals some set union of the elements of $B.$ * A subbasis of $(X,\mathcal O)$ is a subset $S\subseteq\mathcal O$ such that every element $O\in\mathcal O$ equals some set intersection of finitely many elements of $S.$

### Notes

• By convention, the empty union is the empty set and the empty intersection is the whole set $X$. Thus $X$ can be generated by every basis (also just because $X\in\mathcal O$), and $\emptyset$ can be generated by every subbasis.
• The difference between the basis and the subbasis is this: While the basis can generate the whole topology $\mathcal O$, a subbasis, in general, can only generate som subsets of it.
• Since $S\subseteq\mathcal O$, by the second axiom of topology, every intersection $$\bigcap_{\substack{O_i\in S\\\ i\in I\text{ finite}}} O_i$$ is also an element of the topology $\mathcal O.$

### Examples

• The open boxes form a basis of the product topology.
• The open intervals of real numbers $\mathbb R$ with real mid points form a basis of the topological space.

Proofs: 1
Propositions: 2

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
2. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
3. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition