Let (E, r) be a topological space. Let E' = EU {co} with co ¢ E. We define the following topology on E': 9 = (1.1 C E' : (U E r) V (CO E U A CEU is compact)}. • Show that .7 is a indeed a topology. •...

Let (E, r) be a topological space. Let E' = EU {co} with co ¢ E. We define the following topology on E':
9 = (1.1 C E' : (U E r) V (CO E U A CEU is compact)}. • Show that .7 is a indeed a topology. • Show that (E', .9) is compact • A topology r is locally compact when for every x E E them exists a compact r-neighborhood N of x. Show that if (E,r) is locally compact, not compact, then E is dense in (E', g). • Show that if (E, r) is I fausdorff locally compact then {E',.9) is I lausdorff. • Conclude that any locally compact space is an open set in a compact space. • Let f : Lt be a function. Show that f is continuous on (E', .4) if and only if f is continuous on (E, r) and lirnx,00xrt f(x) = f (co). • Show that if h : E —1 X is a continuous injection from (E, r) to X, with r locally compact T2, then there exists a function h' : E'.
Exercise 5. Let X and Y be hvo compact spaces. Let saf be an open covering of X x Y. • Let z E X. Show that there exists a finite subcover A1 AO of el covering the set {x} x Y. • Use the previous exercise to find, for all x E X, an open ncighnborhood Wx of x in X such that Wr x Y U7-1 A,. • Conclude that you can find a subcovering of d which covers X x Y, Le. X x Y is compact.