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IN-CLASS PROBLEM SET JUNE 13, 2013 (1) Find a continuous surjection f : R → {a, b} for each of the following topologies on {a, b}, or explain why no such function exists. In all cases assume R has the standard topology. (a) the discrete topology (b) {∅, {a}, {a, b}} (c) the indiscrete topology (2) Define a relation ∼ = on the set of topological spaces as follows: for X, Y topological spaces, we say X ∼ = Y if there is a homeomorphism f : X → Y . Prove ∼ = is an equivalence relation. (3) Prove that the − δ definition of continuity for functions f : R → R is equivalent to the open set definition (assuming here the standard topology on R). (4) Suppose X, Y, Z are topological spaces and suppose f : X → Y g : Y → Z are continuous functions. Prove g ◦ f is a continuous functon. (5) Suppose f, g : X → R are continuous functions, where X is some topological space and R has the standard topology. Prove f + g is a continuous function. 1