MATH 214 Final, Fall 1995

Mathematics 214, First Semester 1995-96 Final Examination January 29, 1996, 8:00—10:00 Problem 1. 10 pts. Prove that if X x Y is a normal space, then X is a normal space. Problem 2. 10 pts. Let X be a Tichonov space. Prove that each subspace of X is a Tichonov space. Problem 3. 10 pts. Prove that the countable discrete space Z can be embedded into each infinite metric space. Problem 4. 10 pts. Let X be a normal space and Aj, Az and As; be closed pairwise disjoint subsets of X. Use the Urysohn Lemma to obtain a continuous function f :X — R such that f(x) =2 foreachx € Ay, f(a) =5 foreach z€ Ag, and f(z) = —1 for each € As. Problem 5. 15 pts. Let X and Y be topological spaces and A Cc X and B CY. Start from the definition of the product topology on X x Y and give a detailed proof that A x B= AxB. Problem 6. 15 pts. Let X be an infinite set with the co-finite topology (that is a non-empty subset U of X is open iff X — U is a finite set). (a) Prove that X is separable. (b) Prove that X is first countable iff X is countable. (c) Prove that X is compact. Problem 7. 15 pts. Write a proof that each subspace of a separable metric space is separable. Problem 8. 15 pts. Let B denote the family of subsets of [0,2] which consists of: (i) all usual open subsets of [0,2], and (ii) all single point sets {2} where n = 1,2,... Let T denote the topology on [0,2] in which 8 is a basis of open sets. (a) Prove that ({0,2], T) is a regular space. (b) Prove that {0,2],7) is a second countable space. (c) Prove that ({0,2],7) is not a compact space. Bonus problem. 10 pts. Find a subset A of the Euclidean plane R? which is homeomorphic to the space ((0, 2|,T) of Problem 8.