Finite Subcover

Compact Space - Theorems
... A finite union of compact sets is compact ... Any collection of closed subsets of X with the finite intersection property has nonempty intersection ... every cover of the space by members of the sub-base has a finite subcover (Alexander's sub-base theorem) Every net on X has a convergent subnet (see ...

Heine–Borel Theorem - Proof
... Observe first the following if a is a limit point of S, then any finite collection C of open sets, such that each open set U ∈ C is disjoint from some neighborhood VU of a, fails to be a cover of S ... Indeed, the intersection of the finite family of sets VU is a neighborhood W of a in Rn ... Then C ′ is an open cover of S, but any finite subcollection of C ′ has the form of C discussed previously, and thus cannot be an open subcover of S ...

Heine–Borel Theorem - Generalizations
... be an open cover of E, hence there would be a finite subcover (Unk) of E, hence the intersection of the Fnk would be empty this implies that Fn is empty for all n larger ... If E were not compact, there would exist a cover (Ul)l of E having no finite subcover of E ... sequence of balls (Bn) in E with the radius of Bn is 2−n there is no finite subcover (Ul∩Bn)l of Bn Bn+1 ∩ Bn is not empty ...