Difference between revisions of "Category:Topology Theorems"

A

• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell
• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell/Statement
• A continuous map induces a homomorphism between fundamental groups
• A continuous map induces a homomorphism on fundamental groups
• A function is continuous if and only if the pre-image of every basis element is open
• A map is continuous if and only if each point in the domain has an open neighbourhood for which the restriction of the map is continuous on
• A map is continuous if and only if the pre-image of every closed set is closed
• A proper vector subspace of a topological vector space has no interior
• A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
• A set is dense if and only if every non-empty open subset contains a point of it
• A set is open if and only if every point in the set has an open neighbourhood contained within the set
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself
• A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself/Statement
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
• A subspace of a Hausdorff space is a Hausdorff space
• A subspace of a Hausdorff space is Hausdorff
• A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
• A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
• A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
• An open ball contains another open ball centred at each of its points
• An open set is a neighbourhood to all of its points

B

• Basis criterion (topology)
• Basis for a topology

C

• Canonical injection of the subspace topology
• Cauchy-Schwarz inequality for inner product spaces
• Task:Characteristic property of the coproduct topology
• Characteristic property of the disjoint union topology
• Characteristic property of the disjoint union topology/Statement
• Characteristic property of the product topology
• Characteristic property of the product topology/Statement
• Characteristic property of the quotient topology
• Task:Characteristic property of the subspace topology
• Characteristic property of the subspace topology
• Characteristic property of the subspace topology/Statement

C cont.

• Compact-to-Hausdorff theorem
• Compactness/Uniting covers proof
• Composition of continuous maps is continuous
• Continuity definitions are equivalent
• Continuous map/Claim: continuous iff continuous at every point

D

• Discrete metric and topology/Summary

E

• Equivalence of Cauchy sequences/Proof
• Equivalent conditions to a map being a quotient map
• Equivalent conditions to a set being bounded
• Equivalent conditions to a set being bounded/Statement
• Task:Equivalent properties to homeomorphism
• Equivalent statements to a set being dense
• Every continuous map from a non-empty connected space to a discrete space is constant
• Every convergent sequence is Cauchy
• Every lingering sequence has a convergent subsequence
• Every map from a space with the discrete topology is continuous
• Every subspace of a Hausdorff space is Hausdorff

F

• Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map
• For a vector subspace of a topological vector space if there exists a non-empty open set contained in the subspace then the spaces are equal

G

• Given a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself
• Given a topological manifold of dimension 2 or more and points p1, p2 and q where q is neither p1 nor p2 then a path from p1 to p2 is path-homotopic to a path that doesn't go through q
• Given two open balls sharing the same centre but with differing radius then the one defined to have a strictly smaller radius is contained in the other

H

• Homeomorphic topological spaces have isomorphic fundamental groups/Statement
• Homotopy invariance of loop concatenation
• Homotopy invariance of path concatenation
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
• Homotopy of maps is an equivalence relation

I

• If a set is a neighbourhood to all of its points then it is open
• If a subsequence of a Cauchy sequence converges then the Cauchy sequence itself also converges
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
• If the intersection of two open balls is non-empty then for every point in the intersection there is an open ball containing it in the intersection
• If two charts are smoothly compatible with an atlas then they are smothly compatible with each other
• Image of a connected set is connected
• Induced homomorphism on fundamental groups

I cont.

• Intermediate value theorem

L

• Lebesgue number lemma

M

• Metric topology

P

• Passing to the quotient (topology)
• Passing to the quotient (topology)/Statement
• Pasting lemma
• Product topology
• Proof that the fundamental group is actually a group
• Proof that the fundamental group is actually a group/Outline

R

• R^n is a topological vector space

S

• Square lemma (of homotopic paths)

T

• The basis criterion (topology)
• The basis criterion (topology)/Statement
• The canonical injections of the disjoint union topology are topological embeddings
• The closure of a linear subspace of a normed space is a linear subspace
• The composition of continuous maps is continuous
• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
• The image of a compact set is compact
• The image of a connected set is connected
• The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms
• The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms/Statement
• The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group
• The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group/Statement
• The interior of a set in a topological space is equal to the union of all interior points of that set
• The relation of maps being homotopic is an equivalence relation
• The relation of path-homotopy is preserved under composition with continuous maps
• The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
• The stages of a homotopy are continuous
• Topology generated by a basis
• Topology induced by a metric
• Topology theorems

U

• Unique lifting property
• Universal property of the quotient topology
• Urysohn's lemma