Characterization of Continuous Maps in Terms of Preservation of Connected Sets

We prove that if X is locally connected and Y is locally compact Hausdorff, then f: X -> Y is continuous if and only if f has a closed graph and f(C) is connected for any connected subset C of X. This result is not new. We show by examples that the conditions on XContinue reading “Characterization of Continuous Maps in Terms of Preservation of Connected Sets”

Borel’s Proof of the Heine-Borel Theorem

We present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.