Topological Spaces#
- class sage.categories.topological_spaces.TopologicalSpaces(category, *args)#
Bases:
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
- class CartesianProducts(category, *args)#
Bases:
CartesianProductsCategory
- extra_super_categories()#
Implement the fact that a (finite) Cartesian product of topological spaces is a topological space.
EXAMPLES:
sage: from sage.categories.topological_spaces import TopologicalSpaces sage: C = TopologicalSpaces().CartesianProducts() sage: C.extra_super_categories() [Category of topological spaces] sage: C.super_categories() [Category of Cartesian products of sets, Category of topological spaces] sage: C.axioms() frozenset()
- class Compact(base_category)#
Bases:
CategoryWithAxiom
The category of compact topological spaces.
- class CartesianProducts(category, *args)#
Bases:
CartesianProductsCategory
- extra_super_categories()#
Implement the fact that a (finite) Cartesian product of compact topological spaces is compact.
EXAMPLES:
sage: from sage.categories.topological_spaces import TopologicalSpaces sage: C = TopologicalSpaces().Compact().CartesianProducts() sage: C.extra_super_categories() [Category of compact topological spaces] sage: C.super_categories() [Category of Cartesian products of topological spaces, Category of compact topological spaces] sage: C.axioms() frozenset({'Compact'})
- class Connected(base_category)#
Bases:
CategoryWithAxiom
The category of connected topological spaces.
- class CartesianProducts(category, *args)#
Bases:
CartesianProductsCategory
- extra_super_categories()#
Implement the fact that a (finite) Cartesian product of connected topological spaces is connected.
EXAMPLES:
sage: from sage.categories.topological_spaces import TopologicalSpaces sage: C = TopologicalSpaces().Connected().CartesianProducts() sage: C.extra_super_categories() [Category of connected topological spaces] sage: C.super_categories() [Category of Cartesian products of topological spaces, Category of connected topological spaces] sage: C.axioms() frozenset({'Connected'})
- class SubcategoryMethods#
Bases:
object
- Compact()#
Return the subcategory of the compact objects of
self
.EXAMPLES:
sage: Sets().Topological().Compact() Category of compact topological spaces
- Connected()#
Return the full subcategory of the connected objects of
self
.EXAMPLES:
sage: Sets().Topological().Connected() Category of connected topological spaces
- class sage.categories.topological_spaces.TopologicalSpacesCategory(category, *args)#