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finite intersection property
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(Definition)
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A collection
 of subsets of a set  is said to have the finite intersection property, abbreviated f.i.p., if every finite subcollection
 of
 satisifes
 .
The finite intersection property is most often used to give the following equivalent condition for the compactness of a topological space (a proof of which may be found here):
Proposition A topological space  is compact if and only if for every collection
 of closed subsets of  having the finite intersection property,
 .
An important special case of the preceding is that in which
 is a countable collection of non-empty nested sets, i.e., when we have
 . |
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In this case,
 automatically has the finite intersection property, and if each  is a closed subset of a compact topological space, then, by the proposition,
 .
The f.i.p. characterization of compactness may be used to prove a general result on the uncountability of certain compact Hausdorff spaces, and is also used in a proof of Tychonoff's Theorem.
- 1
- J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.
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"finite intersection property" is owned by azdbacks4234. [ full author list (4) | owner history (3) ]
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(view preamble)
See Also: compact, intersection, finite
| Other names: |
finite intersection condition, f.i.c., f.i.p. |
| Also defines: |
finite intersection property |
| Keywords: |
compact, intersection, finite |
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Cross-references: proof of Tychonoff's theorem, Hausdorff spaces, characterization, countable, closed subsets, compact, proof, topological space, finite, subsets, collection
There are 7 references to this entry.
This is version 14 of finite intersection property, born on 2003-04-12, modified 2007-06-22.
Object id is 4178, canonical name is FiniteIntersectionProperty.
Accessed 9564 times total.
Classification:
| AMS MSC: | 54D30 (General topology :: Fairly general properties :: Compactness) |
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Pending Errata and Addenda
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