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A set with a specified family of open subsets.
wp-top-spacenlab-top-spacestacks-topologytopology-without-tearsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Open metric balls generate the canonical topology of a metric space.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-metricnlab-metricwp-top-spacenlab-top-spacetopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Forgetting the structure sheaf and prime-ideal interpretation leaves the underlying Zariski topological space of .
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
stacks-schemes-textstacks-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Choose open subsets satisfying the topology axioms.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-top-spacenlab-top-spacetopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Geometric realization canonically constructs a topological space from a simplicial set ; together with the singular complex it relates the corresponding homotopy theories.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Every uniformity canonically determines a topology by its entourages.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-uniformnlab-uniformwp-top-spacetopology-without-tearsEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Take the smallest -algebra containing every open set. This measurable structure is canonical for the topology.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-borelwp-top-spacetao-measure-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Require every open cover to have a finite subcover.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-compacttopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Require closedness/topology to be detected by compact Hausdorff probes.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-compactly-generatednlab-compactly-generatedhatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Require no separation into two nonempty disjoint open subsets.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-connected-spacenlab-connected-spacetopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Require the identity map to be homotopic to a constant map. A particular contraction is not part of the structure.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-contractiblenlab-contractiblehatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Choose cells and attaching maps satisfying the CW axioms.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-cw-complexnlab-cw-complexhatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Require a countable neighborhood basis at every point.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-first-countabletopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
A homotopy is a continuous one-parameter family of maps.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Homotopy type treats spaces as equivalent when maps in both directions compose to maps homotopic to the corresponding identities, rather than when the spaces are homeomorphic.
Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Supply topological-space structures on the function domain and codomain.
Feed several structures into a construction junction and impose compatibility between them.
wp-continuous-functionnlab-continuous-maptopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Instantiate topological-space structure on the spaces occurring in the map.
Feed several structures into a construction junction and impose compatibility between them.
wp-covering-spacenlab-covering-spacetopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Instantiate compatible topological-space structures on total space , base , and typical fiber .
Feed several structures into a construction junction and impose compatibility between them.
wp-fiber-bundlenlab-fiber-bundletopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Topological space structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-pointed-spacenlab-pointed-spacehatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Topological space structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
stacks-ringed-spacenlab-ringed-spacewp-ringed-spacestacks-schemes-textThis is an authored directed relation from the source endpoint to the target endpoint.
Supply a topology on the same carrier.
Feed several structures into a construction junction and impose compatibility between them.
wp-top-fieldtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the topology.
Feed several structures into a construction junction and impose compatibility between them.
wp-top-grouptopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Topological space structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-topological-semigroupnlab-topological-monoidtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply a topology on the same carrier.
Feed several structures into a construction junction and impose compatibility between them.
stacks-top-ringtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Topological space structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-topological-semigroupnlab-topological-monoidtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the topology on the vector carrier.
Feed several structures into a construction junction and impose compatibility between them.
wp-top-vectorprinceton-functional-analysisThis is an authored directed relation from the source endpoint to the target endpoint.
Require compact neighborhoods under the selected convention.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-locally-compacttopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Require connected neighborhoods to form local bases.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-locally-connectedtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Require local homeomorphisms to open subsets of Euclidean space.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-locally-euclideantopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Require Hausdorffness and second countability, and add an atlas of charts to open subsets of the closed half-space with smooth transition maps.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-manifold-boundarynlab-manifold-boundarylee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Require every two points to be joined by a path.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-path-connectednlab-path-connectedhatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Require the topology to be separable and induced by at least one complete metric.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-polishtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Given a topological space and an equivalence relation, replace points by their equivalence classes and declare open in exactly when its quotient-map preimage is open in .
Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.
topology-without-tearshatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Require a countable basis for the topology.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-second-countabletopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Continuous simplices and alternating face maps form the singular chain complex.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Distinct points must be topologically distinguishable.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-separationtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a subspace .
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-topological-pairhatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Require every connected component to be a singleton.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-totally-disconnectedtopology-without-tearsThis is an authored directed relation from the source endpoint to the target endpoint.
Require a basis consisting of clopen sets.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-zero-dimensionaltopology-without-tears