Properties (50)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
neighborhood
|
| gptkbp:hasExhibition |
Normal Space
Completely Regular Space Metric_Space |
| gptkbp:hasSpecialty |
Any two distinct points can be separated by neighborhoods.
The diagonal is closed in the product space. |
| https://www.w3.org/2000/01/rdf-schema#label |
Hausdorff Space
|
| gptkbp:isCharacterizedBy |
The space is first countable if every point has a countable neighborhood base.
Closed sets contain limit points. Compact subsets are closed. Every convergent sequence has a unique limit. The T2 separation axiom. The intersection of two closed sets is closed. The space is compact if it is closed and bounded. The space is completely regular. The space is first countable. The space is locally compact. The space is metrizable. The space is paracompact. The space is regular. The space is second countable. The space is separable if it contains a countable dense subset. The space is locally path-connected if every point has a neighborhood base of path-connected sets. The space is a completely regular space if points can be separated from closed sets by continuous functions. The space is compact if every open cover has a finite subcover. The space is totally bounded if for every ε>0, it can be covered by finitely many balls of radius ε. The closure of a set is the intersection of all closed sets containing it. The space is a regular space if points can be separated from closed sets. The space is connected if it cannot be divided into two disjoint non-empty open sets. The space is locally connected if every point has a neighborhood base of connected sets. The space is path-connected if any two points can be joined by a continuous path. The space is simply connected if it is path-connected and every loop can be contracted. The union of any collection of closed sets is closed. The space is metrically separable if it is separable and metrizable. The space is quasi-compact if every open cover has a finite subcover. The space is second countable if it has a countable base for its topology. The_space_is_a_Baire_space_if_the_intersection_of_countably_many_dense_open_sets_is_dense. The_space_is_weakly_Hausdorff_if_every_filter_converges_to_at_most_one_point. Every_continuous_image_of_a_compact_Hausdorff_space_is_Hausdorff. Every_compact_Hausdorff_space_is_normal. The_space_is_strongly_Hausdorff_if_every_filter_converges_to_at_most_one_point_in_the_closure. The_space_is_homeomorphic_to_a_closed_subset_of_a_Euclidean_space. The_space_is_compactly_generated_if_it_is_compact_and_Hausdorff. |
| gptkbp:isNear |
Discrete Space
Indiscrete Space |
| gptkbp:isRelatedTo |
Topology
Separation Axioms |
| gptkbp:isUsedIn |
Geometry
Analysis Algebraic_Topology |