Geometry and Topology

Question 1

Properties of a metric

Answer 1

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)

Question 2

Properties of a norm

Answer 2

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)

Question 3

Lift metric

Answer 3

Question 4

Open ball
Closed ball
Sphere

radius r around a

Answer 4

Question 5

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

Answer 5

Question 6

Proof of a Euclidean isometry f

Answer 6

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

Question 7

Euclidean norm

Answer 7

Question 8

Euclidean metric / distance

Answer 8

Question 9

Orthogonal group

Answer 9

Square matrices, where its transpose equals its inverse

Question 10

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

Answer 10

Question 11

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

Answer 11

Question 12

Lipschitz map properties

Answer 12

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

Question 13

Contraction (mapping theorem)

Answer 13

A contraction is a Lipschitz function with constant < 1:

Question 14

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

Answer 14

Question 15

Completeness of a metric space (X,d)

Answer 15

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

Question 16

Cauchy sequence definition

Answer 16

Question 17

Diameter of a (sub)set A

Answer 17

Question 18

Closure of a subset A

Answer 18

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

Question 19

Heine-Borel theorem

Answer 19

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

Question 20

Properties of a topology

Answer 20

A family 𝜏 of subsets of a set X is a topology on X if it satisfies:
* X,∅ 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)

Question 21

Closed sets are…

Answer 21

compliments of open sets

Question 22

Closure of a subset

of a topological space

Answer 22

The smallest closed subset that contains it

Question 23

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

Answer 23

• X and ∅ 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

Question 24

Connected topological space X

Answer 24

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

Question 25

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

Answer 25

`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`

Question 26

Partition

Answer 26

* 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

Question 27

Proof of disconnectedness of a topological space

Answer 27

A topological space `(X,𝜏)` is disconnected if and only if: * ∃ 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`

Question 28

Injective Surjective Bijective | From `X` to `Y`

Answer 28

* 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

Question 29

Path-connected

Answer 29

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

Question 30

Continuous path

Answer 30

A cont. path in a topo. space `(X,𝜏)` is a continuous map 𝛾:[𝑎,𝑏] → 𝑋 , `a

Question 31

Subspace topology on `A`

Answer 31

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

Question 32

Hausdorff space

Answer 32

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

Question 33

Dense

Answer 33

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

Question 34

Metric topology

Answer 34

The metric topology 𝜏,𝑚 is created from the basis 𝐵,𝑚. 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

Question 35

Sunflower metric

Answer 35

Question 36

Interior of a set `A` | and definition of an interior point

Answer 36

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`

Question 37

Homeomorphism between two metric spaces `X` and `Y`

Answer 37

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

Question 38

Cofinite topology

Answer 38

* A family `𝜏` on an infinite set `X`, containing `∅` and all cofinite subsets `U` in `X` * A subset `U` is cofinite if and only if the complement `U^c = X\U` is finite

Question 39

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

Answer 39

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

Question 40

Isometry

Answer 40

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

Question 41

`||u||^2`

Answer 41

`u . u`

Question 42

Euclidean transformation

Answer 42

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`

Question 43

`x . y`

Answer 43

`(x^T)y`