Geometry and Topology Flashcards

1
Q

Properties of a metric

A

A metric is a map d: X x X -> R satisfying:
* d is positive definite: for all x,y in X, d(x,y) >= 0, and d(x,y) = 0 if and only if x = y
* d is symmetric: for all x,y in X, d(x,y) = d(y,x)
* d satisfies the triangle inequality: for all x,y,z in X, d(x,z) <= d(x,y) + d(y,z)

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2
Q

Properties of a norm

A

A normed vector space over a field F, is a vector field V equipped with a map, the norm, ||-||: V -> F satisfying:
* for all x in V, ||x|| >= 0, and ||x|| = 0 if and only if x = 0 is in V
* for all a in F, x in V, ||a x|| = |a| ||x||
* for all x,y in V, ||x + y|| <= ||x|| + ||y|| (triangle inequality)

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3
Q

Lift metric

A
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4
Q

Open ball
Closed ball
Sphere

radius r around a

A
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5
Q

Proof of a continuous map f over metric spaces X,Y

A
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6
Q

Proof of a Euclidean isometry f

A

A Euclidean isometry is a map f: E^n -> E^n which preserves the Euclidean distance

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7
Q

Euclidean norm

A
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8
Q

Euclidean metric / distance

A
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9
Q

Orthogonal group

A

Square matrices, where its transpose equals its inverse

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10
Q

d(1,n,inf) for vectors x,y

A
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11
Q

d(1,n,inf) for functions f,g

A
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12
Q

Lipschitz map properties

A

A map f: X -> Y satisfies the Lipschitz condition if:

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13
Q

Contraction (mapping theorem)

A

A contraction is a Lipschitz function with constant < 1:

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14
Q

Proof of a sequence {x(n)} converging to x

A
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15
Q

Completeness of a metric space (X,d)

A

The metric space is complete if every Cauchy sequence in X converges to a point in X

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16
Q

Cauchy sequence definition

A
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17
Q

Diameter of a (sub)set A

A
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18
Q

Closure of a subset A

A
  • Μ…A is the set of all adherent points of A
  • Μ…A is the intersection of all closed subsets V of X containing A (smallest closed subset containing A)
19
Q

Heine-Borel theorem

A

A subset A of R^n is sequentially compact if and only if it is closed and bounded

20
Q

Properties of a topology

A

A family 𝜏 of subsets of a set X is a topology on X if it satisfies:
* X,βˆ… exist in 𝜏
* the union of any collection of subsets from 𝜏 is in 𝜏
* the intersection of any finite number of subsets from 𝜏 is in 𝜏

Elements of 𝜏 are the open sets of the topological space (X,𝜏)

21
Q

Closed sets are…

A

compliments of open sets

22
Q

Closure of a subset

of a topological space

A

The smallest closed subset that contains it

23
Q

Open and closed sets of a topological space (X,𝜏)

A
  • X and βˆ… are both open and closed
  • The union of any finite family of closed sets is closed
  • The intersection of any family of closed subsets of X is closed
  • The union of any collection of open subsets is open
  • The intersection of any finite number of open subsets is open
24
Q

Connected topological space X

A

X is connected if and only if:
* the only subsets of X which are both open and closed are X and βˆ…
* OR X doesn’t admit a partition

25
Q

Adherent point of a subset A of a topological space (X,𝜏)

A

x is an adherent point of A if and only if:
every non-empty subset U of X containing x, has a non-empty intersection with A

26
Q

Partition

A
  • A partition {A,B} of a topological space (X,𝜏) is a pair of non-empty disjoint subsets, s.t. X = A U B

disjoint: no elements in common

27
Q

Proof of disconnectedness of a topological space

A

A topological space (X,𝜏) is disconnected if and only if:
* βˆƒ a continuous surjective map from X to the two-point discrete space {0,1}
* OR it admits a partition of open disjoint subsets of X

28
Q

Injective
Surjective
Bijective

From X to Y

A
  • Injective: no two different elements of X map to the same element of Y
  • Surjective: every y in Y has at least one x in X that maps to it
  • Bijective: f: X -> Y is both injective and surjective
29
Q

Path-connected

A

A space X is path-connected if any two points of X can be joined by a continuous path, i.e.:

30
Q

Continuous path

A

A cont. path in a topo. space (X,𝜏) is a continuous map 𝛾:[π‘Ž,𝑏] β†’ 𝑋 , a<b,
and the interval [a,b] in R is equipped with the standard (subspace) topology

31
Q

Subspace topology on A

A

For A a subset of the topo. space (X,𝜏):

32
Q

Hausdorff space

A

A topological space (X,𝜏) is Hausdorff if any two distinct points in X have disjoint open neighbourhoods, i.e.:

33
Q

Dense

A

A subset A of a topo. space (X,𝜏) is dense in X if: Μ…A = X

34
Q

Metric topology

A

The metric topology 𝜏,π‘š is created from the basis 𝐡,π‘š.
i.e. a subset U of X in the metric topology is open if and only if U is a union of a family of open balls

35
Q

Sunflower metric

A
36
Q

Interior of a set A

and definition of an interior point

A

The interior of A is the set of all interior points.
x in A is an interior point if and only if there exists an open subset U_x of X s.t. x is in U_x and U_x is a subset of A

37
Q

Homeomorphism between two metric spaces X and Y

A

A homeomorphism of metric spaces is a bijection f: X -> Y such that f and f^(-1) are continuous

38
Q

Cofinite topology

A
  • A family 𝜏 on an infinite set X, containing βˆ… and all cofinite subsets U in X
  • A subset U is cofinite if and only if the complement U^c = X\U is finite
39
Q

How to test for the existence of a norm inducing a metric d

A

d must be translation invariant, i.e. d(x+v,y+v) = d(x,y)

40
Q

Isometry

A

A map f: X -> Y is an isometry (distance-preserving) if:

41
Q

||u||^2

A

u . u

42
Q

Euclidean transformation

A

A Euclidean transformation is a map f_A,t such that f_A,t(x)=Ax+t,
where A is an nxn orthogonal matrix and t is in R^n

43
Q

x . y

A

(x^T)y