O N 93% 1 2:47 Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both infinite X is homeomorphic to Y if and only if X and Y are both finite X is never...

Extracted text: O N 93% 1 2:47 Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both infinite X is homeomorphic to Y if and only if X and Y are both finite X is never homeomorphic to Y X is homeomorphic to Y if and only if X and Y have the same cardinality Which one of the following statements is true? * R with the Euclidean topology and R with the finite closed topology are not homeomorphic None of the choices R with the Euclidean topology and R with the finite closed topology are homeomorphic R with the Euclidean topology and R with the discrete topology are homeomorphic


Extracted text: O N 93% 1 2:47 Let T_us be the usual topology on R and T_II be the lower limit topology generated by the unions of {Ja,b]/ a,bER;asb}. Define f the mapping from (R, T_us) into (R, T_II) by f(x) = 2x and g the mapping from (R, T_II) into (R, T_us) by g(x) = 3x. Then * None of the choices g is a homeomorphism but f is not f and g are both homeomorphisms f is a homeomorphism but g is not Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable}. Then * O (X,T) is connected (X,T) is not connected None of the choices (X,T) is homeomorphic to (X,T1) where T1 is the finite closed topology on X