MATH 417 Midterm: MATH 417 McGill Examf97
15 Feb 2019
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189-354a: a metric space (x, ) is said to be locally compact if for every x x, there exists an (i) rn is locally compact, and open set ux, with x ux and u x compact. Prove that (ii) 1 is not locally compact: let (x, ) be a metric space with at least 2 distinct points. Show that there exists a non constant continuous function x r. if further x is connected, show that x must be uncountable. 3. (a) de ne what is meant by a connected component of a metric space (x, ). If e x is non-empty, open, closed and connected, show that e is a component. (b) if (x, ) is a connected metric space, f : x y is a continuous map, show that f(x) is connected. Show that the circle t = {(x, y) r2 : x2 + y 2 = 1} is not homeomorphic to the closed interval [0, 1].
