A Short and Self-contained Proof of the Insolvability of the Quintic by Radicals

This note contains a short and self-contained proof of the Abel–Ruffini theorem that the quintic equations are not solvable by radicals. Although our proof is based on Galois’s theory, the only technical prerequisites are some familiarity with polynomials and countability.

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”