Topological Spaces¶
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class
sage.categories.topological_spaces.
TopologicalSpaces
(category, *args)¶ Bases:
sage.categories.topological_spaces.TopologicalSpacesCategory
The category of topological spaces.
EXAMPLES:
sage: Sets().Topological() Category of topological spaces sage: Sets().Topological().super_categories() [Category of sets]
The category of topological spaces defines the topological structure, which shall be preserved by morphisms:
sage: Sets().Topological().additional_structure() Category of topological spaces
TESTS:
sage: TestSuite(Sets().Topological()).run()
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class
Compact
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom
The category of compact topological spaces.
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class
TopologicalSpaces.
Connected
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom
The category of connected topological spaces.
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class
TopologicalSpaces.
SubcategoryMethods
¶ -
Compact
()¶ Return the subcategory of the compact objects of
self
.EXAMPLES:
sage: Sets().Topological().Compact() Category of compact topological spaces
TESTS:
sage: TestSuite(Sets().Topological().Compact()).run() sage: Sets().Topological().Compact.__module__ 'sage.categories.topological_spaces'
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Connected
()¶ Return the full subcategory of the connected objects of
self
.EXAMPLES:
sage: Sets().Topological().Connected() Category of connected topological spaces
TESTS:
sage: TestSuite(Sets().Topological().Connected()).run() sage: Sets().Topological().Connected.__module__ 'sage.categories.topological_spaces'
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-
class
-
class
sage.categories.topological_spaces.
TopologicalSpacesCategory
(category, *args)¶ Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()