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3
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'subbasis'
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| Title of object: |
subbasis |
| Canonical Name: |
Subbasis |
| Type: |
Definition |
| Created on: |
2002-01-06 15:50:30-05 |
| Modified on: |
2002-02-18 21:48:27-05 |
| Classification: |
msc:54A99 |
| Keywords: |
topology |
| Synonyms: |
subbasis=subbasic
subbasis=subbasic |
Revision comment (for changes between this and next version):
| Changes for correction #1702 ('typo: \matcal -> \mathcal'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $(X,\mathcal{T})$ be a topological space. A subset $\mathcal{A}\subseteq\matcal{T}$ is said to be a \emph{subbasis} if the collection $\mathcal{B}$ of intersections of finitely many elements of $\mathcal{A}$ is a \PMlinkname{basis}{BasisTopologicalSpace} 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. $\mathcal{T}$ willl then be the smallest topology such that $\mathcal{A}\subseteq\mathcal{T}$. |
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