Topological Spaces

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()
class Compact(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

The category of compact topological spaces.

class TopologicalSpaces.Connected(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom

The category of connected topological spaces.

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'
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'
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()