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- Relatively_compact_subspace abstract "In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact.Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Such a subset may also be called relatively bounded, or pre-compact, although the latter term is also used for a totally bounded subset. (These are equivalent in a complete space.)Some major theorems characterise relatively compact subsets, in particular in function spaces. An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterises relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).The definition of an almost periodic function F at a conceptual level has to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.As a counterexample take any neighbourhood of the particular point of an infinite particular point space. The neighbourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space.".
- Relatively_compact_subspace wikiPageExternalLink v=onepage&q&f=false.
- Relatively_compact_subspace wikiPageID "596600".
- Relatively_compact_subspace wikiPageRevisionID "543779380".
- Relatively_compact_subspace hasPhotoCollection Relatively_compact_subspace.
- Relatively_compact_subspace subject Category:Compactness_(mathematics).
- Relatively_compact_subspace subject Category:Properties_of_topological_spaces.
- Relatively_compact_subspace type Abstraction100002137.
- Relatively_compact_subspace type Possession100032613.
- Relatively_compact_subspace type PropertiesOfTopologicalSpaces.
- Relatively_compact_subspace type Property113244109.
- Relatively_compact_subspace type Relation100031921.
- Relatively_compact_subspace comment "In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact.Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X.".
- Relatively_compact_subspace label "Podprzestrzeń warunkowo zwarta".
- Relatively_compact_subspace label "Relativ kompakte Teilmenge".
- Relatively_compact_subspace label "Relatively compact subspace".
- Relatively_compact_subspace label "Sottospazio relativamente compatto".
- Relatively_compact_subspace label "相対コンパクト部分空間".
- Relatively_compact_subspace sameAs Relativně_kompaktní_množina.
- Relatively_compact_subspace sameAs Relativ_kompakte_Teilmenge.
- Relatively_compact_subspace sameAs Sottospazio_relativamente_compatto.
- Relatively_compact_subspace sameAs 相対コンパクト部分空間.
- Relatively_compact_subspace sameAs 상대_콤팩트_부분공간.
- Relatively_compact_subspace sameAs Podprzestrzeń_warunkowo_zwarta.
- Relatively_compact_subspace sameAs m.02tt59.
- Relatively_compact_subspace sameAs Q610232.
- Relatively_compact_subspace sameAs Q610232.
- Relatively_compact_subspace sameAs Relatively_compact_subspace.
- Relatively_compact_subspace wasDerivedFrom Relatively_compact_subspace?oldid=543779380.
- Relatively_compact_subspace isPrimaryTopicOf Relatively_compact_subspace.