Topology Problems

I. In class, we considered a pair of functions: /: R ➔ R, /(:r:) = sin(:r:); and g: R ➔ R2 , g(x) = (cos(x),sin(x)). (a) Compute 1- 1 ( ½)- (b) Compute g- 1 ((1,0)). 2. How many distinct topologies are there on the set X = { a, b, c}? For the purposes of this problem, consider two topologies T and T' distinct if, as collections of subsets of X, T 'I- T' • For example, T = {0, {a}, X} and T' = {a, {b}, X} are c.li1;;ti11ct topologies. 3. Let T and T' be topologies on X. Prove or give a counterexample: ( a) T n T' is also a topology on X. (b) TU T' is also a topology on X. • 5. Let /3 = { la, b) I a, b E R, a < b}. {a) Prove that /3 defines a basis for a topology on IR. (b) If T is the u.suaJ topology on JR and T' is the topology on JR induced by /3, how do T and T' compare·! Explain. 6. Consider the function f : R --; JR defined by f(x) = {X X < 1 x+l xl Show that f is not continuous if JR is given the usual topology, but becomes continuous using the topology T' from Problem f>. 7. Suppose (X, T) is a topological space and.that /3 is a basis for T. Prove that if Y c X has the subspace topology Ty inherited from X, then {Jy = { B n Y I B E f3} is a basis for Ty. 8. Let R have the usual topology. (a) Prove that the subspace topology on Z inherited from R coincides with the discrete the topology on Z. 9. Let f : X ➔ Y be a function between topological spaces X a.nd Y. Prove that / is continuous if and only if 1-1 ( C) C X is closed whenever C C Y is closed.