MATH 4565 Study Guide - Final Guide: Retract, Homeomorphism, Contractible Space
9 Jan 2019
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Final exam: let f : x y be a continuous, bijective map. X is compact and y is hausdor , then f is a homeomorphism. Show that both assumptions are necessary for the theorem to hold. Hausdor , but f is not a homeomorphism: let x = {(x, y) r2 | x q or y q} be the set of all points in the plane with at least one rational coordinate. Show that x, with the induced topology, is a path-connected space: let f and g be paths in r2 \ {0}. Show that f is homotopic to g: consider the unit circle s 1 = {(x, y) r2 | x2 + y2 = 1}. Let f : s 1 s 1 be the map de ned by f (x, y) = ( x, y). What is the degree of f : let f : s 1 s 1 be a continuous map.
