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
-
class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.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:
sage.categories.category_with_axiom.CategoryWithAxiom
The category of compact topological spaces.
-
class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.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
-
class
Connected
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom
The category of connected topological spaces.
-
class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.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
-
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
-
class
sage.categories.topological_spaces.
TopologicalSpacesCategory
(category, *args)¶ Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory