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