# Topological Spaces#

class sage.categories.topological_spaces.TopologicalSpaces(category, *args)[source]#

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 *
Category of topological spaces
```
class CartesianProducts(category, *args)[source]#
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]#

The category of compact topological spaces.

class CartesianProducts(category, *args)[source]#
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]#

The category of connected topological spaces.

class CartesianProducts(category, *args)[source]#
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`

Compact()[source]#

Return the subcategory of the compact objects of `self`.

EXAMPLES:

```sage: Sets().Topological().Compact()
Category of compact topological spaces
```
```>>> from sage.all import *
>>> Sets().Topological().Compact()
Category of compact topological spaces
```
Connected()[source]#

Return the full subcategory of the connected objects of `self`.

EXAMPLES:

```sage: Sets().Topological().Connected()
Category of connected topological spaces
```
```>>> from sage.all import *
>>> Sets().Topological().Connected()
Category of connected topological spaces
```
class sage.categories.topological_spaces.TopologicalSpacesCategory(category, *args)[source]#