# Part A Metric Spaces and Complex Analysis

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1. Definition 1.3. (v^k)k∈N converges to w ∈ R^n if
2. Definition 1.3.1: If f : R^n → R and a ∈ R^n then we say that f is continuous at a
3. Define distance function
4. define metric space
• a pair (X, d) of a set and a distance function d: X × X → R≥0 satisfying the axioms for a distance function. If the distance function is clear from context, we may, for convenience,
• simply write X rather than (X, d).
5. Definition 2.4. Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y
is said to be continuous at a ∈ X if
• for any e > 0 there is a δ > 0 such that for any x ∈ X with dX(a, x) < δ we have dY (f(x), f(a)) < e. We say f is continuous if it
• is continuous at every a ∈ X.
6. If (xn)n≥1 is a sequence in X, and a ∈ X, then we say (xn)n≥1 converges to a
if,
for any e > 0 there is an N ∈ N such that for all n ≥ N we have dX(xn, a) < e.
7. A function f : X → Y is said to be uniformly continuous if
• for any e > 0, there exists a δ > 0 such that for all x1, x2 ∈ X with dX(x1, x2) < δ
• we have dY (f(x1), f(x2)) < e.
8. give three common metrics on R^n
9. define a norm
10. what is
11. define an open ball and a closed ball
12. define open, a neighbourhood and a topology
13. define an interior
14. define a topology, topological space and continuity of a function between two topological spaces
15. define closed
16. define a limit point and an isolated point
17. define closure
18. define the boundary of a subset of a metric space
19. define an isometry
20. define homeomorphism, homeomorphic
21. what is a Cauchy sequence
22. define a complete metric space
23. weierstrauss M-test
24. define Lipschitz map, contraction
25. CMT
26. define disconnected, connected
27. define the connected component
28. state the IVT
29. define path-connected
30. define concatenation, opposite path
31. define path-component of a metric space
 Author: Nat1234 ID: 334924 Card Set: Part A Metric Spaces and Complex Analysis Updated: 2017-10-23 22:49:42 Tags: maths Folders: Description: part a metric spaces and complex analysis Show Answers: