MAT-4710 Midterm: MATH 4710 App State Fall2014 Test2 answer key
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Be sure to show your work: (12 points) an open question. (a) using the standard basis for r (real numbers with lower limit topology), show i = (0, 1) is open. There are two main approaches to this problem. Either we could show that i is the union of intervals of the form [a, b) or we could show that every element of i belongs to some interval [a, b) i. Since i is the union of (basic) open sets, it"s open. Proof #2: let x i = (0, 1). Obviously x [x, 1) and [x, 1) (0, 1) (since x > 0). So every element of i has a (basic open) neighborhood lying in i. Suppose (a, b) is a neighborhood of 1. So b/2 > 1 and thus b/2 6 j. So every open interval containing 1 spills outside of j.
