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- Trivial_topology abstract "In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space. Intuitively, this has the consequence that all points of the space are \"lumped together\" and cannot be distinguished by topological means; it belongs to a pseudometric space in which the distance between any two points is zero.The trivial topology is the topology with the least possible number of open sets, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space X with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space.Other properties of an indiscrete space X—many of which are quite unusual—include: The only closed sets are the empty set and X. The only possible basis of X is {X}. If X has more than one point, then since it is not T0, it does not satisfy any of the higher T axioms either. In particular, it is not a Hausdorff space. Not being Hausdorff, X is not an order topology, nor is it metrizable. X is, however, regular, completely regular, normal, and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X. X is compact and therefore paracompact, Lindelöf, and locally compact. Every function whose domain is a topological space and codomain X is continuous. X is path-connected and so connected. X is second-countable, and therefore is first-countable, separable and Lindelöf. All subspaces of X have the trivial topology. All quotient spaces of X have the trivial topology Arbitrary products of trivial topological spaces, with either the product topology or box topology, have the trivial topology. All sequences in X converge to every point of X. In particular, every sequence has a convergent subsequence (the whole sequence), thus X is sequentially compact. The interior of every set except X is empty. The closure of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense, a property that characterizes trivial topological spaces. As a result of this, the closure of every open subset U of X is either ∅ (if U = ∅) or X (otherwise). In particular, the closure of every open subset of X is again an open set, and therefore X is extremally disconnected. If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \\ S is still a limit point of S. X is a Baire space. Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.Let Top be the category of topological spaces with continuous maps and Set be the category of sets with functions. If F : Top → Set is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and G : Set → Top is the functor that puts the trivial topology on a given set, then G is right adjoint to F. (The functor H : Set → Top that puts the discrete topology on a given set is left adjoint to F.)".
- Trivial_topology wikiPageID "382447".
- Trivial_topology wikiPageLength "5020".
- Trivial_topology wikiPageOutDegree "71".
- Trivial_topology wikiPageRevisionID "651331293".
- Trivial_topology wikiPageWikiLink 0_(number).
- Trivial_topology wikiPageWikiLink 1_(number).
- Trivial_topology wikiPageWikiLink Adjoint_functors.
- Trivial_topology wikiPageWikiLink Baire_space.
- Trivial_topology wikiPageWikiLink Base_(topology).
- Trivial_topology wikiPageWikiLink Box_topology.
- Trivial_topology wikiPageWikiLink Cardinality.
- Trivial_topology wikiPageWikiLink Category:General_topology.
- Trivial_topology wikiPageWikiLink Category:Topological_spaces.
- Trivial_topology wikiPageWikiLink Category:Topology.
- Trivial_topology wikiPageWikiLink Category_of_sets.
- Trivial_topology wikiPageWikiLink Category_of_topological_spaces.
- Trivial_topology wikiPageWikiLink Closed_set.
- Trivial_topology wikiPageWikiLink Closure_(topology).
- Trivial_topology wikiPageWikiLink Codomain.
- Trivial_topology wikiPageWikiLink Compact_space.
- Trivial_topology wikiPageWikiLink Connected_space.
- Trivial_topology wikiPageWikiLink Continuous_function.
- Trivial_topology wikiPageWikiLink Counterexamples_in_Topology.
- Trivial_topology wikiPageWikiLink Dense_set.
- Trivial_topology wikiPageWikiLink Discrete_space.
- Trivial_topology wikiPageWikiLink Domain_of_a_function.
- Trivial_topology wikiPageWikiLink Dover_Publications.
- Trivial_topology wikiPageWikiLink Empty_set.
- Trivial_topology wikiPageWikiLink Extremally_disconnected_space.
- Trivial_topology wikiPageWikiLink First-countable_space.
- Trivial_topology wikiPageWikiLink Forgetful_functor.
- Trivial_topology wikiPageWikiLink Function_(mathematics).
- Trivial_topology wikiPageWikiLink Functor.
- Trivial_topology wikiPageWikiLink Hausdorff_space.
- Trivial_topology wikiPageWikiLink Homeomorphism.
- Trivial_topology wikiPageWikiLink If_and_only_if.
- Trivial_topology wikiPageWikiLink Interior_(topology).
- Trivial_topology wikiPageWikiLink Kolmogorov_space.
- Trivial_topology wikiPageWikiLink Limit_(mathematics).
- Trivial_topology wikiPageWikiLink Limit_point.
- Trivial_topology wikiPageWikiLink Lindelöf_space.
- Trivial_topology wikiPageWikiLink Locally_compact_space.
- Trivial_topology wikiPageWikiLink Metric_(mathematics).
- Trivial_topology wikiPageWikiLink Metrization_theorem.
- Trivial_topology wikiPageWikiLink Normal_space.
- Trivial_topology wikiPageWikiLink Open_set.
- Trivial_topology wikiPageWikiLink Order_topology.
- Trivial_topology wikiPageWikiLink Paracompact_space.
- Trivial_topology wikiPageWikiLink Product_topology.
- Trivial_topology wikiPageWikiLink Pseudometric_space.
- Trivial_topology wikiPageWikiLink Quotient_space_(topology).
- Trivial_topology wikiPageWikiLink Regular_space.
- Trivial_topology wikiPageWikiLink Second-countable_space.
- Trivial_topology wikiPageWikiLink Separable_space.
- Trivial_topology wikiPageWikiLink Separation_axiom.
- Trivial_topology wikiPageWikiLink Sequence.
- Trivial_topology wikiPageWikiLink Sequentially_compact_space.
- Trivial_topology wikiPageWikiLink Singleton_(mathematics).
- Trivial_topology wikiPageWikiLink Springer_Science+Business_Media.
- Trivial_topology wikiPageWikiLink Subspace_topology.
- Trivial_topology wikiPageWikiLink Topological_indistinguishability.
- Trivial_topology wikiPageWikiLink Topological_space.
- Trivial_topology wikiPageWikiLink Topology.
- Trivial_topology wikiPageWikiLink Triviality_(mathematics).
- Trivial_topology wikiPageWikiLink Tychonoff_space.
- Trivial_topology wikiPageWikiLink Uniform_space.
- Trivial_topology wikiPageWikiLinkText "Trivial topology".
- Trivial_topology wikiPageWikiLinkText "indiscrete topology".
- Trivial_topology wikiPageWikiLinkText "indiscrete".
- Trivial_topology wikiPageWikiLinkText "trivial topology".
- Trivial_topology wikiPageWikiLinkText "trivial".
- Trivial_topology wikiPageUsesTemplate Template:Citation.
- Trivial_topology subject Category:General_topology.
- Trivial_topology subject Category:Topological_spaces.
- Trivial_topology subject Category:Topology.
- Trivial_topology type Field.
- Trivial_topology type Space.
- Trivial_topology comment "In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space.".
- Trivial_topology label "Trivial topology".
- Trivial_topology sameAs Q1418327.
- Trivial_topology sameAs Triviale_Topologie.
- Trivial_topology sameAs Topología_trivial.
- Trivial_topology sameAs توپولوژی_بدیهی.
- Trivial_topology sameAs Topologie_grossière.
- Trivial_topology sameAs טופולוגיה_טריוויאלית.
- Trivial_topology sameAs Indiszkrét_topológia.
- Trivial_topology sameAs Topologia_banale.
- Trivial_topology sameAs 密着空間.
- Trivial_topology sameAs 비이산_공간.
- Trivial_topology sameAs Przestrzeń_antydyskretna.
- Trivial_topology sameAs Topologia_grosseira.
- Trivial_topology sameAs m.021sc0.
- Trivial_topology sameAs Тривиальная_топология.
- Trivial_topology sameAs Тривијална_топологија.
- Trivial_topology sameAs Антидискретна_топологія.
- Trivial_topology sameAs Q1418327.
- Trivial_topology sameAs 密着拓扑.
- Trivial_topology wasDerivedFrom Trivial_topology?oldid=651331293.
- Trivial_topology isPrimaryTopicOf Trivial_topology.