Topological Spaces

class sage.categories.topological_spaces.TopologicalSpaces(category, *args)

Bases: TopologicalSpacesCategory

The category of topological spaces.

EXAMPLES:

Sage

sage: Sets().Topological()
Category of topological spaces
sage: Sets().Topological().super_categories()
[Category of sets]

Python

>>> from sage.all import *
>>> Sets().Topological()
Category of topological spaces
>>> Sets().Topological().super_categories()
[Category of sets]

The category of topological spaces defines the topological structure, which shall be preserved by morphisms:

Sage

sage: Sets().Topological().additional_structure()
Category of topological spaces

Python

>>> from sage.all import *
>>> 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

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()

Python

>>> from sage.all import *
>>> from sage.categories.topological_spaces import TopologicalSpaces
>>> C = TopologicalSpaces().CartesianProducts()
>>> C.extra_super_categories()
[Category of topological spaces]
>>> C.super_categories()
[Category of Cartesian products of sets, Category of topological spaces]
>>> 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

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'})

Python

>>> from sage.all import *
>>> from sage.categories.topological_spaces import TopologicalSpaces
>>> C = TopologicalSpaces().Compact().CartesianProducts()
>>> C.extra_super_categories()
[Category of compact topological spaces]
>>> C.super_categories()
[Category of Cartesian products of topological spaces,
 Category of compact topological spaces]
>>> 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

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'})

Python

>>> from sage.all import *
>>> from sage.categories.topological_spaces import TopologicalSpaces
>>> C = TopologicalSpaces().Connected().CartesianProducts()
>>> C.extra_super_categories()
[Category of connected topological spaces]
>>> C.super_categories()
[Category of Cartesian products of topological spaces,
 Category of connected topological spaces]
>>> C.axioms()
frozenset({'Connected'})

class SubcategoryMethods

Bases: object

Compact()

Return the subcategory of the compact objects of self.

EXAMPLES:

Sage

sage: Sets().Topological().Compact()
Category of compact topological spaces

Python

>>> from sage.all import *
>>> Sets().Topological().Compact()
Category of compact topological spaces

Connected()

Return the full subcategory of the connected objects of self.

EXAMPLES:

Sage

sage: Sets().Topological().Connected()
Category of connected topological spaces

Python

>>> from sage.all import *
>>> Sets().Topological().Connected()
Category of connected topological spaces

class sage.categories.topological_spaces.TopologicalSpacesCategory(category, *args)

Bases: RegressiveCovariantConstructionCategory