Difference between revisions of "Category:Topology"

H

• ► Homotopy Theory‎ (2 C, 70 P)

M

• ► Metric Space‎ (3 C, 144 P)

T

• ► Topological Manifolds‎ (1 C, 18 P)

T cont.

• ► Topology Definitions‎ (270 P)
• ► Topology Notes‎ (5 P)
• ► Topology Theorems‎ (1 C, 102 P)

T cont.

• ► Topology Theorems, lemmas and corollaries‎ (102 P)

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
• Adjunction
• Adjunction (topology)
• 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
• Basis criterion (topology)
• Basis for a topology
• Boundary (topology)
• Bounded
• Bounded (linear map)
• Bounded (sequence)
• Bounded (set)
• Bounded linear map
• Bounded sequence
• Bounded set
• Box topology

C

• C(I,X)
• C(X,Y)
• Canonical injection of the subspace topology
• Canonical projection of the product topology
• Canonical projections of the product topology
• Cauchy sequence
• Cauchy sequence/Definition
• Cauchy sequence/Short definition
• 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 product topology/Statement/Diagram
• 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
• Circle
• Closed interval
• Closed map
• Closed n-cell
• Closed set
• Closed sets
• Closure (set, topology)
• Closure (topology)
• Closure of a set in a topological space
• Closure, interior and boundary
• Coarser topology
• Compact
• Compact (topology)

C cont.

• Compact-to-Hausdorff theorem
• Compactness
• Compactness/Uniting covers proof
• Complete metric space
• Completeness
• Composition of continuous maps is continuous
• Concatenation of loops and paths (homotopy)
• Concatenation of paths and loops (homotopy)
• Cone
• Cone (topology)
• Connected (topology)
• Notes:Connected space
• Connected subset (topology)
• Connected topological space
• Constant loop
• Constant loop based at
• Constant loop based at a point
• Continuity
• Continuity and non-surjective functions
• Continuity definitions are equivalent
• Task:Continuity types
• Continuous
• Continuous function
• Continuous map
• Continuous map/Claim: continuous iff continuous at every point
• Continuous maps
• Contractible topological space
• Convergence (sequence)
• Convergence of a sequence
• Convergent sequence
• Converges (sequence)
• Coproduct
• Coproduct (topology)
• Cover
• Covering
• Covering map (topology)
• Covering map (topology)/Definition
• Covering space
• Covering space (topology)
• CW-complex
• Notes:CW-Complex

D

• Deformation retract
• Deformation retraction
• Deformation retraction/Definition
• Dense
• Dense set
• Differentiation
• Disconnected (topology)
• Disconnected (topology)/Definition
• Disconnected subset (topology)
• Disconnected topological space
• Discrete metric
• Discrete metric and topology
• Discrete metric and topology/Metric space definition
• Discrete metric and topology/Summary
• Discrete topology
• Disjoint union
• Disjoint union (topology)
• Disjoint union space (topology)
• Disjoint union topological space
• Disjoint union topology

E

• End point preserving homotopic
• End-point-preseriving homotopic
• End-point-preserving homotopic paths
• Equivalence of Cauchy sequences
• Equivalence of Cauchy sequences/Definition
• Equivalence of Cauchy sequences/Proof

E cont.

• 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
• Euclidean distance
• Euclidean metric
• Even covering (topology)
• Evenly covered (topology)
• Evenly covered by a continuous map
• Evenly covers (topology)
• 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
• Finer topology
• Finite complement topology
• First homotopy group
• 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
• Fundamental group homomorphism induced by a continuous map
• Fundamental groups

G

• Notes:Generalising the limit
• Geometrically independent set
• 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
• Graph
• Graph (topological manifold)

H

• Hausdorff
• Hausdorff space
• Homeomorphic
• Homeomorphic topological spaces have isomorphic fundamental groups
• Homeomorphic topological spaces have isomorphic fundamental groups/Statement
• Homeomorphism
• Homeomorphism (topology)
• Homomorphism
• Homomorphism (topology)
• Homotopic
• Homotopic maps
• Homotopic paths
• Homotopy
• Homotopy class
• Homotopy concatenation
• Homotopy equivalence
• Homotopy equivalent topological space
• Homotopy equivalent topological spaces
• 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
• HomotopyPage

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 compact set is compact
• Image of a connected set is connected
• Indiscrete topology
• Induced homomorphism on fundamental groups
• Template:Inner product and Hilbert spaces navbox
• Inner-product space
• Interior
• Interior (topology)