is there a generalization of the bolzano weirstrasss theorem

is there a generalization of the bolzano weirstrasss theorem involving closed sets? If so state it, if not explain why

Solution

Bolzano Weierstrass theorem states that every bounded subset of R as a convergent subsequence.

This cannot be generalized to closed sets.

For example , consider the set Z of integers in R. Z is closed .. Consider the sequence x[n]=n. It cannot have any convergent subsequence , as it is unbounded.