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METRIC SPACES TOPOLOGY EXAM QUESTIONS AND ANSWERS
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METRIC SPACES TOPOLOGY EXAMQUESTIONS AND ANSWERS
Definition 5.1.1: What is a metric? What is a metric space? - ANSWER A
metric on a set X is a map d: X×X→R that satisfies the following properties for
all x,y,z ∈ X
(i) Positive definiteness: d(x, y) ≥ 0, with d(x, y) = 0 iff x = y
(ii) Symmetry: d(x, y) = d(y, x)
(iii) Triangle inequality: d(x, y) ≤ d(x, z) + d(z, y)
The pair (X, d) is called a metric space (pg. 180)
Explain the relationship between inner product spaces, normed spaces, and
metric spaces - ANSWER An inner product space induces a norm ||x|| = √<x,
x> to produce a normed space which in turn induces a metric d(x, y) = ||x - y|| to
produce a metric space. So inner product spaces are normed spaces are metric
spaces.
Thus metric spaces are more general than normed spaces are more general than
inner product spaces (class)
Example 5.1.2: Explain why the 2-norm on F^n is a metric - ANSWER d(c, y)
= ||x - y||₂
Unless specified, this is always the metric we us on F^n
(pg. 180)
What is the Euclidean metric? - ANSWER The Euclidean Metric on F^n is
given by the 2-norm, that is,
d(x, y) = ||x-y||₂
The 2-norm of their difference
(pg. 180)
Given any normed linear space (V, ||*||), what is the natural metric to use? -
ANSWER d(x, y) = ||x - y||
The norm of the difference
(pg. 181)
,What metric do we typically use for C([a, b]; R)? - ANSWER For p ∈ [1, ∞]
we have
d^p(f, g) = (∫_{a}^{b}|f(t)-g(t)|^p)^(1/p) for p ∈[1, ∞)
d^p(f, g) = sup_{t∈[a, b]}(|f(t) - g(t)|) for p = ∞
(pg. 181)
What is the discrete metric? - ANSWER The discrete metric on X is
d(x, y) = 0 if x=y
d(x, y) = 1 if x≠y
Thus no distinct points are close together-they are always the same distance
apart
(pg. 181)
What is the p-metric? - ANSWER Let ((Xi, di))_{i=1}^n be a collection of
metric spaces, and let X = X1xX2...Xn be the Cartesian product. For any two
points x = (x1, ..., xn), y = (y1, ..., yn), define
d^p(x, y) = (∑_{i=1}^n di(xi, yi)^p)^(1/p) for p ∈[1, ∞)
d^p(x, y) = sup_i(di(xi, yi)) for p = ∞
This defines a metric on X called the p-metric.
(pg. 181)
Example 5.1.3: Explain
(i) why any norm ||*|| on a vector space induces a natural metric
(ii) fixme
(iii) the discrete metric
(iv) the p-metric - ANSWER (pg. 181)
Example 5.1.4: How to use a metric to create a new metric in which no two
points are farther apart than 1 - ANSWER Let (x, d) be a metric space. We can
create a new metric on X:
p(x, y) = d(x, y)/(1+d(x, y))
where no two points are farther apart than 1.(pg. 181)
Remark: 5.1.5: Every norm induces a metric space, but not every metric space is
a normed space - ANSWER (pg. 182)
, Definition 5.1.6: What is an open ball in the metric space (X, d)? - ANSWER
For each point x₀ and r >0, define the open ball with center x₀ and radius r > 0 to
be the set
B(x₀, r) = {x∈X| d(x, x₀) < r}
(pg. 182)
Definition 5.1.7: Let (X, d) be a metric space. What is a neighborhood of a point
x in X? What is an interior point of a subset E⊂X? What is E°? - ANSWER A
subset E ⊂ X is a neighborhood of a point x∈X if there exists an open ball B(x,
r) ⊂ E. In this case we say that x is an interior of E. We write E° to denote the
set of interior points of E (pg. 182)
Definition 5.1.8: What is an open set? - ANSWER A subset E ⊂ X is an open
set if every point x ∈ E is an interior point of E (pg. 182)
Example 5.1.9: Explain why X and ∅ are open sets - ANSWER X is open
since B(x, r) ⊂ X for all x∈X and for all r >0. ∅ is open vacuously: every point
in ∅ satisfies the condition because there are no points in ∅ (pg. 182)
Example 5.1.10: Explain why not all open balls are spherical in R^3 -
ANSWER (pg. 182)
Theorem 5.1.12: Balls as defined in 5.1.8 are open sets. - ANSWER If y ∈
B(x, r) for some x∈X, then B(y, r-ε) ⊂ B(x, r), where ε = d(x, y)
Prove this by taking z∈B(y, r-ε) and showing z∈B(x, r)
See figure 5.1 (pg. 183)
Theorem 5.1.14: The union of any collection of open sets is ... the intersection
of any ... is ... - ANSWER The union of any collection of open sets is open,
and the intersection of any finite collection of open sets is open.
FIXME: practice this proof (pg. 184)
Theorem 5.1.16: Properties of any subset E of X - ANSWER (i) (E°)° = E°,
and hence E° is open
(ii) If G is an open subset of E, then G⊂ E°
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