Let (E, r) be a topological space. Let E’ = EU {co} with co ¢ E. We define… 1 answer below »

Let (E, r) be a topological illimitableness. Let E' = EU {co} delay co ¢ E. We designate the aftercited topology on E':
9 = (1.1 C E' : (U E r) V (CO E U A CEU is amalgamate)}. • Show that .7 is a truly a topology. • Show that (E', .9) is amalgamate • A topology r is discreetly amalgamate when for every x E E them exists a amalgamate r-neighborhood N of x. Show that if (E,r) is discreetly amalgamate, not amalgamate, then E is condensed in (E', g). • Show that if (E, r) is I fausdorff discreetly amalgamate then {E',.9) is I lausdorff. • Conclude that any discreetly amalgamate illimitableness is an notorious set in a amalgamate illimitableness. • Let f : Lt be a office. Show that f is true on (E', .4) if and singly if f is true on (E, r) and lirnx,00xrt f(x) = f (co). • Show that if h : E —1 X is a true introduction from (E, r) to X, delay r discreetly amalgamate T2, then there exists a office h' : E'.
Exercise 5. Let X and Y be hvo amalgamate illimitablenesss. Let saf be an notorious crust of X x Y. • Let z E X. Show that there exists a limited subcover A1 AO of el crust the set {x} x Y. • Use the preceding drill to ascertain, for all x E X, an notorious ncighnborhood Wx of x in X such that Wr x Y U7-1 A,. • Conclude that you can ascertain a subcrust of d which covers X x Y, Le. X x Y is amalgamate.