Quiz 2 Key
MA 366
Name:
1. A metric on X is a function d from X × X to R, with the properties
that for any x, y, z in X, we have
(a) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y.
(b) d(x, y) = d(y, x).
(c) d(x, z) ≤ d(x, y) + d(y, z).
2. Let X be a metric space with metric “d”. Let E ⊂ X, and p ∈ X.
Define the following terms or phrases:
(a) A neighborhood of p is the set {x ∈ X; d(p, x) < r}, for some fixed
r > 0.
(b) A limit point for E is a point p such that every neighborhood of
p contains some point q ∈ E, with q ̸= p.
(c) E is open if, for every x ∈ E there is some neighborhood of x
entirely contained in E.
(d) E is compact if every open covering of E (union ∪α Gα of open
sets that contains E) has a finite subcover (a finite collection of
the Gα contain E).
3. First, Q is not open, since every real number is a limit point of Q (by
virtue of the fact
√ that Q is dense in R), and there are reals that are not
rational, e.g., 2. Also, since every open interval contains rationals
(by density) no point of Q is an interior point, so Q is not open.
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