Prove either one of the following (not both):
1. A naive topological space X is compact if and only if every sequence of points in X has a convergent subsequence.
2. A naive topological space X is compact if and only if every infinite subset of X has a limit point in X.
If you want, you might try to explain:
3. Every sequence of points in X has a convergent subsequence if and only if every infinite subset of X has a limit point in X.
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John H. Palmieri, Department of Mathematics, University of Notre Dame, John.H.Palmieri.2@nd.edu