Topological Closures of Manifold Subsets

ManifoldSubsetClosure implements the topological closure of a manifold subset in the topology of the manifold.

class sage.manifolds.subsets.closure.ManifoldSubsetClosure(subset, name=None, latex_name=None)

Bases: ManifoldSubset

Topological closure of a manifold subset in the topology of the manifold.

INPUT:

• subset – a ManifoldSubset

• name – (default: computed from the name of the subset) string; name (symbol) given to the closure

• latex_name – string (default: None); LaTeX symbol to denote the subset. If none is provided, it is set to name.

EXAMPLES:

Sage

sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D', coord_def={c_cart: x^2+y^2<1}); D
Open subset D of the 2-dimensional topological manifold R^2
sage: cl_D = D.closure()
sage: cl_D
Topological closure cl_D of the Open subset D of the 2-dimensional
 topological manifold R^2
sage: latex(cl_D)
\mathop{\mathrm{cl}}(D)
sage: type(cl_D)
<class 'sage.manifolds.subsets.closure.ManifoldSubsetClosure_with_category'>
sage: cl_D.category()
Category of subobjects of sets

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'R^2', structure='topological')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2
>>> D = M.open_subset('D', coord_def={c_cart: x**Integer(2)+y**Integer(2)<Integer(1)}); D
Open subset D of the 2-dimensional topological manifold R^2
>>> cl_D = D.closure()
>>> cl_D
Topological closure cl_D of the Open subset D of the 2-dimensional
 topological manifold R^2
>>> latex(cl_D)
\mathop{\mathrm{cl}}(D)
>>> type(cl_D)
<class 'sage.manifolds.subsets.closure.ManifoldSubsetClosure_with_category'>
>>> cl_D.category()
Category of subobjects of sets

The closure of the subset \(D\) is a subset of every closed superset of \(D\):

Sage

sage: S = D.superset('S')
sage: S.declare_closed()
sage: cl_D.is_subset(S)
True

Python

>>> from sage.all import *
>>> S = D.superset('S')
>>> S.declare_closed()
>>> cl_D.is_subset(S)
True

is_closed()

Return if self is a closed set.

This implementation of the method always returns True.

EXAMPLES:

Sage

sage: from sage.manifolds.subsets.closure import ManifoldSubsetClosure
sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D', coord_def={c_cart: x^2+y^2<1}); D
Open subset D of the 2-dimensional topological manifold R^2
sage: cl_D = D.closure(); cl_D  # indirect doctest
Topological closure cl_D of the Open subset D of the 2-dimensional topological manifold R^2
sage: cl_D.is_closed()
True

Python

>>> from sage.all import *
>>> from sage.manifolds.subsets.closure import ManifoldSubsetClosure
>>> M = Manifold(Integer(2), 'R^2', structure='topological')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2
>>> D = M.open_subset('D', coord_def={c_cart: x**Integer(2)+y**Integer(2)<Integer(1)}); D
Open subset D of the 2-dimensional topological manifold R^2
>>> cl_D = D.closure(); cl_D  # indirect doctest
Topological closure cl_D of the Open subset D of the 2-dimensional topological manifold R^2
>>> cl_D.is_closed()
True