Geometry and Topology Flashcards
Properties of a metric
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)
Properties of a norm
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)
Lift metric
Open ball
Closed ball
Sphere
radius r around a
Proof of a continuous map f over metric spaces X,Y
Proof of a Euclidean isometry f
A Euclidean isometry is a map f: E^n -> E^n which preserves the Euclidean distance
Euclidean norm
Euclidean metric / distance
Orthogonal group
Square matrices, where its transpose equals its inverse
d(1,n,inf) for vectors x,y
d(1,n,inf) for functions f,g
Lipschitz map properties
A map f: X -> Y satisfies the Lipschitz condition if:
Contraction (mapping theorem)
A contraction is a Lipschitz function with constant < 1:
Proof of a sequence {x(n)} converging to x
Completeness of a metric space (X,d)
The metric space is complete if every Cauchy sequence in X converges to a point in X
Cauchy sequence definition
Diameter of a (sub)set A
Closure of a subset A
- Μ
Ais the set of all adherent points ofA - Μ
Ais the intersection of all closed subsetsVofXcontainingA(smallest closed subset containingA)
Heine-Borel theorem
A subset A of R^n is sequentially compact if and only if it is closed and bounded
Properties of a topology
A family π of subsets of a set X is a topology on X if it satisfies:
* X,β
exist in π
* the union of any collection of subsets from π is in π
* the intersection of any finite number of subsets from π is in π
Elements of π are the open sets of the topological space (X,π)
Closed sets areβ¦
compliments of open sets
Closure of a subset
of a topological space
The smallest closed subset that contains it
Open and closed sets of a topological space (X,π)
-
Xandβare both open and closed - The union of any finite family of closed sets is closed
- The intersection of any family of closed subsets of
Xis closed - The union of any collection of open subsets is open
- The intersection of any finite number of open subsets is open
Connected topological space X
X is connected if and only if:
* the only subsets of X which are both open and closed are X and β
* OR X doesnβt admit a partition
Adherent point of a subset A of a topological space (X,π)
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
Partition
- A partition
{A,B}of a topological space(X,π)is a pair of non-empty disjoint subsets, s.t.X = A U B
disjoint: no elements in common
Proof of disconnectedness of a topological space
A topological space (X,π) is disconnected if and only if:
* β 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
Injective
Surjective
Bijective
From X to Y
- Injective: no two different elements of
Xmap to the same element ofY - Surjective: every
yinYhas at least onexinXthat maps to it - Bijective:
f: X -> Yis both injective and surjective
Path-connected
A space X is path-connected if any two points of X can be joined by a continuous path, i.e.:
Continuous path
A cont. path in a topo. space (X,π) is a continuous map πΎ:[π,π] β π , a<b,
and the interval [a,b] in R is equipped with the standard (subspace) topology
Subspace topology on A
For A a subset of the topo. space (X,π):
Hausdorff space
A topological space (X,π) is Hausdorff if any two distinct points in X have disjoint open neighbourhoods, i.e.:
Dense
A subset A of a topo. space (X,π) is dense in X if: Μ
A = X
Metric topology
The metric topology π,π is created from the basis π΅,π.
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
Sunflower metric
Interior of a set A
and definition of an interior point
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
Homeomorphism between two metric spaces X and Y
A homeomorphism of metric spaces is a bijection f: X -> Y such that f and f^(-1) are continuous
Cofinite topology
- A family
πon an infinite setX, containingβand all cofinite subsetsUinX - A subset
Uis cofinite if and only if the complementU^c = X\Uis finite
How to test for the existence of a norm inducing a metric d
d must be translation invariant, i.e. d(x+v,y+v) = d(x,y)
Isometry
A map f: X -> Y is an isometry (distance-preserving) if:
||u||^2
u . u
Euclidean transformation
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
x . y
(x^T)y
