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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 $\mathcal{A}\subseteq\mathcal{T}$
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"subbasis" is owned by evin290. [ full author list (2) | owner history (1) ]
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Cross-references: basis, generated by, finite, conversely, 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 7396 times total.
Classification:
| AMS MSC: | 54A99 (General topology :: Generalities :: Miscellaneous) |
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Pending Errata and Addenda
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