Let (X, τ) be a topological space. Show that every sequence in X has a convergent subsequence.

Question

In the previous parts, I was given that (X,τ) is compact. I have proved that every closed set of X is compact, and that every infinite subset of X has a limit point.

I was struggling with these question and found this thread

choose the correct following option?

This says that every sequence in a compact space need not have a convergent subsequence? I am very confused, as I don't think there is an error in the question, and I don't think they wouldn't phrase it like this if they were looking for a counterexample.

Is there something I have misunderstood or am missing?

Any help is great appreciated! Also any hints on how to start the proof would be a huge help!

Answer 1

There is nothing wrong with the statement. In a compact metric space every sequence has convergent subsequence but thus is not always true in compact topological spaces. Every sequence in a compact topological space has convergent subnet but not necessarily a convergent subsequence.

For additional information please search for 'sequential compactness' in Wikipedia.