Question

Q: 6 State and prove finite intersection property.​

Answer 1

Answer:

Let {\displaystyle X}X be a set and let {\displaystyle {\mathcal {A}}=\{A_{i}\}_{i\in I}}{\displaystyle {\mathcal {A}}=\{A_{i}\}_{i\in I}} a family of subsets of {\displaystyle X}X indexed by an arbitrary set {\displaystyle I}I. Then the collection {\displaystyle {\mathcal {A}}}{\mathcal {A}} has the finite intersection property (FIP), if any finite subcollection {\displaystyle J\subseteq I}J\subseteq I has non-empty intersection, that is {\displaystyle \bigcap _{i\in J}A_{i}}{\displaystyle \bigcap _{i\in J}A_{i}} is a non-empty subset of {\displaystyle X.}X.

Answer 2

Answer:

In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A is infinite.

A centered system of sets is a collection of sets with the finite intersection property.