Chapter 7 Flashcards
Why study metric spaces?
To see what we can learn about a set of points if all we know is the distance between two points.
What is a metric? (Formal definition)
A function p on X x X such that:
1. p(x,y) >= 0, with p(x,y)=0 iff x=y
2. p(x,y) = p(y,x)
3. p(x,z) <= p(x,y) + p(y,z) (triangle inequality)
What is a metric? (Informal definition)
A function of X x X where you stick in two points and the distance between them is returned. That also satisfies the three axioms.
What is a metric space?
A space formed by a nonempty set and a metric for that space. So that the distance between every point in the set is known.
What is the most basic thing that makes up a metric space?
Open and closed balls of a fixed radius.
Open ball? Quantitative and simple explanation?
B(y,r) is equivalent to {x: p(y,x) < r}
A ball with radius r > 0 and center y in X to be all the points x in X whose distance to y is less than r.
Closed ball? Quantitative and simple explanation?
B^bar (y,r) is equivalent to {x: p(y,x) <= r}
A ball with radius r > 0 and center y in X to be all the points x in X whose distance to y is less than or equal to r.
What is the main difference between an open versus a closed ball?
The open ball does not include its boundary points, while the closed ball does include its boundary points.
Closed set?
A set E is closed if its complement is open. Or if it contains all of its limit/boundary points.
Open set?
A set E is open if for every point x in E there exists a radius r > 0 small enough that B(x,r) is a subset of E. Or in other words, there exists a ball for every point in E.
Convergence? (Formal definition)
We say that a sequence {x_i}^infinity _ i=1 subset of X converges to x* if lim(i to infinity) p(x_i,x*)=0.
Convergence? (Informal definition)
We say that a sequence y converges to an arbitrary point x if as we move further along the sequence, the distance between the sequence term and x approaches 0.
Inverse image? (Formal definition)
For f: X –> Y, we define f^-1 (A) equivalent to {x | f(x) \in A subset Y}.
Inverse image? (Informal definition)
We say that f: X –> Y is the inverse image of A under f if for all x in X there exists a f(x) in A which is a subset of Y.
Continuity definition 1? (Formal)
A function mapping from one metric space X to another Y, (X could equal Y), is said to be continuous if for every open set E subset Y, the inverse image of E is open in X.
Continuity definition 2? (Formal)
A function mapping from one metric space X to another metric space Y, (X could equal Y), is said to be continuous if: x_i –> x^* implies f(x_i) –> f(x^*)
Continuity definition 1*? (Formal)
A function mapping omega subset X to another metric space Y, (X could equal Y), is said to be continuous if for every open set E subset Y, the inverse image f^-1 (E) = omega intersect F for some open F subset X.
Continuity definition 1*? (Informal)
A function f from a subset omega of X –> Y is continuous if, for any open set E in Y, the part of omega that maps into E is the intersection of omega with some open set in X.
Cauchy sequences? (Formal)
A sequence {x_i}^infinity_(i=1) subset X is a cauchy sequence if, for every epsilon > 0, there is an N(epsilon) such that i,j > N(epsilon) implies that p(xi,xj) < epsilon.
Cauchy sequences? (Informal)
A sequence whose terms become close to each other as the sequence progresses.
Completeness?
We say that a metric space (p, X) is complete if every cauchy sequence has a limit in X.
Continuity 3? (Formal)
A function mapping from omega subset X to another metric space Y, (X could equal Y), is said to be continuous if, for every point x in omega and epsilon > 0, there is a delta(x,epsilon) >0 such that f(B(x,delta(x,epsilon)) intersect omega) subset B(f(x), epsilon).
When in R^n is a set compact?
If and only if it is closed and bounded.
Compact sets? (Informal)
A set K subset X is compact if every open cover of K contains a finite subcover.
