Math 145 first exam, S19
Throughout this exam, X is a metric space with metric d, and R is the set of real numbers
with the usual metric.
1. Give a definition, or something equivalent to the definition, for the following.
(a) Open.
(b) Convergent sequence.
(c) Compact.
2. Prove that if A and B are subsets of X then int(A) ∩ int(B) ⊆ int(A ∩B).
3. Fix a point x0 ∈ X. Prove that {x ∈ X | d(x, x0) ≤ 1} is closed.
4. Prove that every convergent sequence is Cauchy.
5. State your favorite metric on the product space X ×X. Use it to prove that if U is an
open subset of X then U × U is an open subset of X ×X.
6. Fix a point x0 ∈ X. Prove that the function f : X → R given by f(x) = d(x, x0) is
continuous.

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