- For any basis , the union of the sets in is equal to . Phrased differently, for any element , there exists a basis set such that .
- For any two sets , given an element , then there exists another set such that .
This definition is very useful for comparing different topologies. In particular, we have the following theorem:
Theorem: In a space , given two topologies and , then iff for any basis element and any element , there exists a basis element such that .
A sub-basis of is a collection of sets whose union is . The collection of intersection of sets in forms a basis on .
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