MATH 214 Final, Fall 2007

Math 214 Final Exam, 23/01/2008 1. (a) (10 points) Let X = {1, 2, 3}. Show that the discrete topology is the only topology on X which is T1. (b) (10 points) Let X, Y be topological spaces, and let Y be a T2-space. As- sume f : X → Y is one-one, onto and continuous. Prove that X is T2, too. (c) (20 points) Let (X, τ) be a T2-space and A ⊂ X. Show the following property: x ∈ A ⇐⇒ x ∈ A, or each nhood of x contains infinitely many points of A. (1) Hint : Use the following well-known criterion, (in (2) below, X is a topological space, A ⊂ X, and x ∈ X): x ∈ A ⇐⇒ U ∩ A ̸= ∅ for every open nhood of x. (2) 2. This exercise illustrates that, in general, sequences alone are inadequate to characterize topologies. – Let X be an uncountable set. (a) (10 points) Show that if X is equipped with the discrete topology, then (∗) every convergent sequence is constant from some term on, and such a sequence converges to that constant value. (b) (10 points) Show that property (*) also holds in the following topology τ = {U ⊂ X : X \ U is at most countable }.