advertisement

Topology M.S. exam. 1. Prove or disprove: If U is an open set then U = int(Ū ). Here int(U ) denotes the interior and Ū denotes the closure of the set U . 2. Show that every compact Hausdorff space is normal. 3. A relation ∼ on X is an equivalence relation if 1. x ∼ x for all x ∈ X; 2. if x ∼ y then y ∼ x for all x, y ∈ X; 3. if x ∼ y and y ∼ z then x ∼ z for all x, y, z ∈ X. Let X be a connected space. Suppose that ∼ is an equivalence relation on X with the property that, for each x ∈ X there is an open neighborhood Ux of x such that x∼y for all y ∈ Ux . Prove that x ∼ y for all x, y ∈ X.