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CHAPTER 4 – BASIC PROPERTIES METRIC SPACES TEST QUESTIONS AND ANSWERS
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CHAPTER 4 – BASIC PROPERTIESMETRIC SPACES TEST QUESTIONS
AND ANSWERS
4.1 - Basic Properties of Metric Spaces
4.1 - Basic Properties of Metric Spaces
What does the real number form?
A chain in which any set bounded from above has a least upper bound, and
dually, any set bounded from below has a greater lower bound.
A systematic notation for the least upper bound
sup. If A is a set of real numbers bounded from above, we write sup A for the
least upper bound of A.
Notation for when A is not bounded from above
sup A = ∞
Notation for greatest lower bound
inf
What can we use instead of sup and inf if A is finite?
max and min instead of sup and inf
Definition - Metric space
A metric space is a set M equipped with a real-valued function D(a,b) defined
for all a,b∈M so as to satisfy:
I. D(a,a)=0 for all a.
II. D(a,b)>0 for a≠b.
, III. D(a,b)=D(b,a).
IV. D(a,c) ≤ D(a,b) + D(b,c) (triangular inequality)
How can we think about the function D(a,b)?
As the distance between a and b.
Example of a metric space
The set of all real numbers, with D(a,b) = |a-b|
Any set of real numbers
is a metric space relative to the distance function D(a,b) = |a-b|
Definition - The Euclidean plane
The set of all ordered pairs of real numbers.
Definition - Euclidean distance
Let M be the usual Euclidean plane: the set of all ordered pairs of real numbers.
Two typical points of M are u = (x₁,y₁), v = (x₂,y₂). The Euclidean distance is
given by
D(u,v) = √(x₁-x₂)²+(y₁-y₂)²
Ath. að rótin nær yfir allt!
An arbitrary set equipped with a trivial distance function.
If M is any set, take D(a,a)=0 and D(a,b)=1 for a≠b in M.
Theorem 26 - For any points a,b,c in a metric space we have
|D(a,c) -D(b,c)|≤ D(a,b)
Definition - Diameter
The diameter of a metric M is sup(D(a,b)), taken over all a,b∈M.
4.2 - Open sets
4.2 - Open sets
Definition- Isometry
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