In topology, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty.
This is trivially satisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection, e.g. the nested sequence (0, 1/n). These are not the only possibilities however. For example, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion with digit 0 in the i'th decimal place, then any finite intersection is nonempty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all Xi for i≥1 is empty, since no element of (0, 1) has all zero digits.
The finite intersection property is useful in formulating an alternative definition of compactness. In particular, a space is compact if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself. This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers.
In topology, the finiteintersectionproperty is a property of a collection of subsets of a set X.
This is trivially satisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection, e.g.
The finiteintersectionproperty is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets satisfying the finiteintersectionproperty has nonempty intersection itself.
To actually prove Tychonoff's theorem, we use the definition of compactness based on the FIP, by taking an FIP collection A of sets, and showing that the intersection over closures of elements of A is nonempty.
But then these basis elements intersect every element of D, and so x is a limit point of each element of D, and so is in the closure of each element of D.
Another proof uses the Alexander subbase theorem, and yet another proof follows trivially from the properties of nets on product spaces, in particular that a net converges in a product space iff each coordinate converges and the fact that compactness can be expressed in terms of nets.
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