Topological Spaces¶
- class sage.categories.topological_spaces.TopologicalSpaces(category, *args)[source]¶
- Bases: - TopologicalSpacesCategory- The category of topological spaces. - EXAMPLES: - sage: Sets().Topological() Category of topological spaces sage: Sets().Topological().super_categories() [Category of sets] - >>> 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: Sets().Topological().additional_structure() Category of topological spaces - >>> from sage.all import * >>> Sets().Topological().additional_structure() Category of topological spaces - class CartesianProducts(category, *args)[source]¶
- Bases: - CartesianProductsCategory- extra_super_categories()[source]¶
- 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() - >>> 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)[source]¶
- Bases: - CategoryWithAxiom- The category of compact topological spaces. - class CartesianProducts(category, *args)[source]¶
- Bases: - CartesianProductsCategory- extra_super_categories()[source]¶
- 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'}) - >>> 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)[source]¶
- Bases: - CategoryWithAxiom- The category of connected topological spaces. - class CartesianProducts(category, *args)[source]¶
- Bases: - CartesianProductsCategory- extra_super_categories()[source]¶
- 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'}) - >>> 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[source]¶
- Bases: - object