Sequential compactness is a property of a topological space, which means that every sequence in the space has a convergent subsequence.
Sequential compactness is a weaker form of compactness, which means that not every sequence in a compact space has a convergent subsequence, while in a sequentially compact space, every sequence has a convergent subsequence.
In a metric space, sequential compactness is equivalent to compactness, which is a key property in the continuity of functions. This means that in a sequentially compact space, continuous functions will preserve compactness.
No, not every metric space is sequentially compact. For a metric space to be sequentially compact, it must also be complete and totally bounded.
Some examples of sequentially compact spaces include closed and bounded intervals in the real numbers, the Cantor set, and finite sets with the discrete topology.