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| 1 | Compact |
| 1 | Topological Space |
| 1 | T Space |
| 1 | Closed Set |
| 1 | Proof |
| 1 | Collection |
| 1 | Finite |
| 1 | Countable |
| 1 | Subset |
| 2 | Characterization |
| 7 | Proof Of Tychonoffs Theorem |
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Finite Intersection Property
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):
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
 . |
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.
Bibliography
- 1
- J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.