Metric Spaces Point-Set Topology Flashcards
(42 cards)
1
Q
Convergence
A
- Let (x_n)n=m,inf be a sequence of real numbers, and let x be another real number. Then (x_n)n=m,if converges to x iff lim(n->inf) d(x_n,x) = 0.
2
Q
Metric Space
A
- A space X of objects (called points) + a distance function / metric d: X x X -> [0, inf)
- Each pair of points x, y in X are “mapped” to a non-negative real-number d(x,y) >= 0.
3
Q
Metric Space Axioms
A
- For all x in X, d(x,x) = 0.
- For any distinct x, y in X, we have d(x,y) > 0.
- For any x, y in X, d(x,y) = d(y,x).
- For any x, y, z in X, d(x,z) <= d(x,y) + d(y,z).
4
Q
Induced Metric Spaces
A
- (X, d) a metric space, Y contained in X.
- Restrict d: X x X -> [0, inf) to subset Y x Y of X x X to create restricted metric function d|Y x Y : Y x Y -> [0, inf).
- This is a metric on Y induced by the metric d on X.
5
Q
Euclidean Metric (d_l2)
A
- R^n x R^n for d_l2((x_1,…,x_n),(y_1,…,y_n)) = (sum(i=1 to n) (x_i - y_i)^2)^0.5
6
Q
Taxicab Metric (d_l1)
A
- R^n x R^n for d_l1((x_1,…,x_n),(y_1,…,y_n)) = (sum(i=1 to n) (x_i - y_i))
7
Q
d_l1 and d_l2 Relationship
A
d_l2(x,y) <= d_l1(x,y) <= n^0.5 * d_l2(x,y)
8
Q
Sup Norm Metric (d_linf)
A
- R^n x R^n for d_linf((x_1,…,x_n),(y_1,…,y_n)) = sup(|x_i - y_i|: 1 <= i <= n)
9
Q
d_l2 and d_linf Relationship
A
(1 / (n^.5)) * d_l2(x,y) <= d_linf(x,y) <= d_l2(x,y)
10
Q
Discrete Metric (d_disc)
A
- d_disc(x,y) = {0 when x=y, 1 when x /= y}
11
Q
Convergence of Sequences in Metric Spaces
A
- (x_n)n=m,inf converges to x wrt metric d iff lim(n->inf) d(x_n,x) = 0.
- This means for all e > 0, there exists an N >= m : d(x_n,x) < e for all n >= N.
12
Q
Ball
A
- B_(X, d)(x_0,r) := {x in X, r > - : d(x,x_0) < r}
- An open set.
13
Q
Interior Point
A
- (X, d) metric space, E contained in X, x_0 in X.
- x_0 an interior point of E if there is an r > 0 : B(x_0,r) contained in E.
14
Q
Exterior Point
A
- x_0 an exterior point of E if there is an r > 0 : B(x_0,r) intersect E = empty set.
15
Q
Boundary Point
A
- x_0 is neither an interior point nor a boundary point.
16
Q
Boundary
A
- The set of all boundary points.
17
Q
Adherent Point
A
- (X,d) metric space, E a subset of X, x_0 in X.
- x_0 an adherent point of E if for all r > 0, B(x_0, r) intersect E /= empty set (nonempty intersection).
18
Q
Closure
A
- The set of all adherent points.
- Denoted E_bar.
19
Q
Equivalence of Points Proposition
A
- (X,d) metric space, E a subset of X, x_0 in X.
- x_0 an adherent point of E iff x_0 is either an interior or boundary point of E iff there exists a sequence (x_n)n=1, inf in E -> x_0 wrt metric d.
20
Q
Closure, Interior, Boundary, Exterior Relationship
A
- (X,d) metric space, E a subset of X.
E_bar = int(E) U dE = X\ext(E)
21
Q
Closed
A
- (X,d) metric space, E a subset of X.
- E closed if E contains all its boundary point (dE c= E).
- E closed iff E contains all its adherent points.
- E closed iff for all convergent sequences (x_n)n=m,inf in E, lim(n->inf) x_n lies in E.
22
Q
Open
A
- (X,d) metric space, E a subset of X.
- E is open if it contains none of its boundary points (dE intersect E = empty set).
- E is open iff E = int(E)
- E is open iff for all x in E, there is an r > 0 : B(x,r) c= E.
23
Q
Closed Ball
A
- The set {x in X : d(x,x_0) < r}
24
Q
Singleton Set
A
- (X,d) metric space.
- Any singleton set {x_0}, where x_0 in X, is automatically closed.
25
Intersection of Finite Open Sets
- Intersection of Finite Open Sets is Open.
26
Union of Finite Closed Sets
- Union of Finite Closed Sets is Closed.
27
Union of Collection of Open Sets
- Union of Collection of Open sets is Open.
28
Intersection of Collection of Closed Sets
- Intersection of Collection of Closed Sets is Closed.
29
Largest Open Set
- (X,d) metric space.
| - If E is any subset of X, int(E) is the largest open set in E.
30
Smallest Closed Set
- E_bar is smallest closed set which contains E.
31
Relatively Open
- (X,d) metric space, Y a subset of X, E a subset of Y.
| - E is relatively open wrt Y if E is open in metric subspace (Y,d|YxY).
32
Relatively Closed
- (X,d) metric space, Y a subset of X, E a subset of Y.
| - E is relatively closed wrt Y if E is closed in metric subspace (Y,d|YxY).
33
Subsequence
- A sequence (x_nj)n=1,inf of elements from the sequence (x_n)n=m,inf.
34
Sequence, Subsequence Convergence
- Let (x_n)n=m,inf be a sequence in (X,d) which converges to x_0. Then every subsequence (x_nj)j=1,inf of that sequence converges to x_0.
35
Limit Points
- (x_n)n=m,inf sequence in (X,d) metric space, L in X.
- L a limit point of (x_n)n=m,inf iff for every N >= m, e > 0, there exists n >= N : d(x_n,L) < e.
- L is a limit point of (x_n)n=m,inf <=> exists a subsequence (x_nj)j=1,inf of (x_n)n=m,inf which converges to L.
36
Cauchy Sequence
- (x_n)n=m,inf sequence in (X,d).
- (x_n)n=m,inf is a cauchy sequence iff for all e > 0, there exists an N >= m : d(x_j,x_k) < e for all j,k >= N.
- Convergent sequences are Cauchy sequences.
37
Complete
- (X,d) metric space is complete iff every Cauchy sequence in (X,d) converges in (X,d).
38
Compact
- (X,d) metric space is compact iff every sequence in (X,d) has at least one convergent subsequence.
- Subset Y of (X,d) is compact if the subspace (Y,d|YxY) is compact.
39
Bounded
- (X,d) metric space, Y a subset of X.
| - Y bounded iff there exists B(x,r) in X which contains Y.
40
Compact, Boundedness and Completeness
- A compact subset Y of a metric space (X,d) is both complete and bounded
41
Open Cover Theorem
- Let Y be a compact subset of a metric space (X,d).
- Then every open cover of Y (meaning you can cover Y by open subsets) has a finite subcover (you only need a finite number of those subsets to cover Y).
42
Union of Compact Sets
- The union of compact sets is also compact.
