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subbasis (Definition)

Let $ (X,\mathcal{T})$ be a topological space. A subset $ \mathcal{A}\subseteq\mathcal{T}$ is said to be a subbasis if the collection $ \mathcal{B}$ of intersections of finitely many elements of $ \mathcal{A}$ is a basis for $ \mathcal{T}$.

Conversely, given an arbitrary collection $ \mathcal{A}$ of subsets of $ X$, a topology can be formed by first taking the collection $ \mathcal{B}$ of finite intersections of members of $ \mathcal{A}$ and then taking the topology $ \mathcal{T}$ generated by $ \mathcal{B}$ as basis. $ \mathcal{T}$ will then be the smallest topology such that $ \mathcal{A}\subseteq\mathcal{T}$.



"subbasis" is owned by evin290. [ full author list (2) | owner history (1) ]
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See Also: basis, basis (topology)

Other names:  subbasic, subbasic
Keywords:  topology
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Cross-references: basis, generated by, finite, intersections, collection, subset, topological space
There are 15 references to this entry.

This is version 5 of subbasis, born on 2002-01-06, modified 2004-06-22.
Object id is 1409, canonical name is Subbasis.
Accessed 6009 times total.

Classification:
AMS MSC54A99 (General topology :: Generalities :: Miscellaneous)

Pending Errata and Addenda
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Subbasis: Missing condition? by kompik on 2005-09-10 11:45:13
I think that condition that the union of A is the whole X should be required in the second paragraph.
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