Subsets of Topological Manifolds

The class ManifoldSubset implements generic subsets of a topological manifold. Open subsets are implemented by the class TopologicalManifold (since an open subset of a manifold is a manifold by itself), which inherits from ManifoldSubset.

AUTHORS:

• Eric Gourgoulhon, Michal Bejger (2013-2015): initial version

• Travis Scrimshaw (2015): review tweaks; removal of facade parents

REFERENCES:

• [Lee2011]

EXAMPLES:

Two subsets on a manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: a = M.subset('A'); a
Subset A of the 2-dimensional topological manifold M
sage: b = M.subset('B'); b
Subset B of the 2-dimensional topological manifold M
sage: M.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset B of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M]

The intersection of the two subsets:

sage: c = a.intersection(b); c
Subset A_inter_B of the 2-dimensional topological manifold M

Their union:

sage: d = a.union(b); d
Subset A_union_B of the 2-dimensional topological manifold M

Lists of subsets after the above operations:

sage: M.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset A_inter_B of the 2-dimensional topological manifold M,
 Subset A_union_B of the 2-dimensional topological manifold M,
 Subset B of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M]
sage: a.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset A_inter_B of the 2-dimensional topological manifold M]
sage: c.list_of_subsets()
[Subset A_inter_B of the 2-dimensional topological manifold M]
sage: d.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset A_inter_B of the 2-dimensional topological manifold M,
 Subset A_union_B of the 2-dimensional topological manifold M,
 Subset B of the 2-dimensional topological manifold M]

class sage.manifolds.subset.ManifoldSubset(manifold, name, latex_name=None, category=None)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

Subset of a topological manifold.

The class ManifoldSubset inherits from the generic class Parent. The corresponding element class is ManifoldPoint.

Note that open subsets are not implemented directly by this class, but by the derived class TopologicalManifold (an open subset of a topological manifold being itself a topological manifold).

INPUT:

• manifold – topological manifold on which the subset is defined

• name – string; name (symbol) given to the subset

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

• category – (default: None) to specify the category; if None, the category for generic subsets is used

EXAMPLES:

A subset of a manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: from sage.manifolds.subset import ManifoldSubset
sage: A = ManifoldSubset(M, 'A', latex_name=r'\mathcal{A}')
sage: A
Subset A of the 2-dimensional topological manifold M
sage: latex(A)
\mathcal{A}
sage: A.is_subset(M)
True

Instead of importing ManifoldSubset in the global namespace, it is recommended to use the method subset() to create a new subset:

sage: B = M.subset('B', latex_name=r'\mathcal{B}'); B
Subset B of the 2-dimensional topological manifold M
sage: M.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset B of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M]

The manifold is itself a subset:

sage: isinstance(M, ManifoldSubset)
True
sage: M in M.subsets()
True

Instances of ManifoldSubset are parents:

sage: isinstance(A, Parent)
True
sage: A.category()
Category of subobjects of sets
sage: p = A.an_element(); p
Point on the 2-dimensional topological manifold M
sage: p.parent()
Subset A of the 2-dimensional topological manifold M
sage: p in A
True
sage: p in M
True

Element

alias of sage.manifolds.point.ManifoldPoint

ambient()

Return the ambient manifold of self.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: A = M.subset('A')
sage: A.manifold()
2-dimensional topological manifold M
sage: A.manifold() is M
True
sage: B = A.subset('B')
sage: B.manifold() is M
True

An alias is ambient:

sage: A.ambient() is A.manifold()
True

declare_union(dom1, dom2)

Declare that the current subset is the union of two subsets.

Suppose \(U\) is the current subset, then this method declares that \(U\)

\[U = U_1 \cup U_2,\]

where \(U_1 \subset U\) and \(U_2 \subset U\).

INPUT:

• dom1 – the subset \(U_1\)

• dom2 – the subset \(U_2\)

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: A = M.subset('A')
sage: B = M.subset('B')
sage: M.declare_union(A, B)
sage: A.union(B)
2-dimensional topological manifold M

get_subset(name)

Get a subset by its name.

The subset must have been previously created by the method subset() (or open_subset())

INPUT:

• name – (string) name of the subset

OUTPUT:

• instance of ManifoldSubset (or of the derived class TopologicalManifold for an open subset) representing the subset whose name is name

EXAMPLES:

sage: M = Manifold(4, 'M', structure='topological')
sage: A = M.subset('A')
sage: B = A.subset('B')
sage: U = M.open_subset('U')
sage: M.list_of_subsets()
[Subset A of the 4-dimensional topological manifold M,
 Subset B of the 4-dimensional topological manifold M,
 4-dimensional topological manifold M,
 Open subset U of the 4-dimensional topological manifold M]
sage: M.get_subset('A')
Subset A of the 4-dimensional topological manifold M
sage: M.get_subset('A') is A
True
sage: M.get_subset('B') is B
True
sage: A.get_subset('B') is B
True
sage: M.get_subset('U')
Open subset U of the 4-dimensional topological manifold M
sage: M.get_subset('U') is U
True

intersection(other, name=None, latex_name=None)

Return the intersection of the current subset with another subset.

INPUT:

• other – another subset of the same manifold

• name – (default: None) name given to the intersection in the case the latter has to be created; the default is self._name inter other._name

• latex_name – (default: None) LaTeX symbol to denote the intersection in the case the latter has to be created; the default is built upon the symbol \(\cap\)

OUTPUT:

• instance of ManifoldSubset representing the subset that is the intersection of the current subset with other

EXAMPLES:

Intersection of two subsets:

sage: M = Manifold(2, 'M', structure='topological')
sage: a = M.subset('A')
sage: b = M.subset('B')
sage: c = a.intersection(b); c
Subset A_inter_B of the 2-dimensional topological manifold M
sage: a.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset A_inter_B of the 2-dimensional topological manifold M]
sage: b.list_of_subsets()
[Subset A_inter_B of the 2-dimensional topological manifold M,
 Subset B of the 2-dimensional topological manifold M]
sage: c._supersets  # random (set output)
{Subset B of the 2-dimensional topological manifold M,
 Subset A_inter_B of the 2-dimensional topological manifold M,
 Subset A of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M}

Some checks:

sage: (a.intersection(b)).is_subset(a)
True
sage: (a.intersection(b)).is_subset(a)
True
sage: a.intersection(b) is b.intersection(a)
True
sage: a.intersection(a.intersection(b)) is a.intersection(b)
True
sage: (a.intersection(b)).intersection(a) is a.intersection(b)
True
sage: M.intersection(a) is a
True
sage: a.intersection(M) is a
True

is_open()

Return if self is an open set.

This method always returns False, since open subsets must be constructed as instances of the subclass TopologicalManifold (which redefines is_open)

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: A = M.subset('A')
sage: A.is_open()
False

is_subset(other)

Return True if and only if self is included in other.

EXAMPLES:

Subsets on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: a = M.subset('A')
sage: b = a.subset('B')
sage: c = M.subset('C')
sage: a.is_subset(M)
True
sage: b.is_subset(a)
True
sage: b.is_subset(M)
True
sage: a.is_subset(b)
False
sage: c.is_subset(a)
False

lift(p)

Return the lift of p to the ambient manifold of self.

INPUT:

• p – point of the subset

OUTPUT:

• the same point, considered as a point of the ambient manifold

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: A = M.open_subset('A', coord_def={X: x>0})
sage: p = A((1, -2)); p
Point on the 2-dimensional topological manifold M
sage: p.parent()
Open subset A of the 2-dimensional topological manifold M
sage: q = A.lift(p); q
Point on the 2-dimensional topological manifold M
sage: q.parent()
2-dimensional topological manifold M
sage: q.coord()
(1, -2)
sage: (p == q) and (q == p)
True

list_of_subsets()

Return the list of subsets that have been defined on the current subset.

The list is sorted by the alphabetical names of the subsets.

OUTPUT:

• a list containing all the subsets that have been defined on the current subset

Note

To get the subsets as a Python set, used the method subsets() instead.

EXAMPLES:

Subsets of a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: U = M.open_subset('U')
sage: V = M.subset('V')
sage: M.list_of_subsets()
[2-dimensional topological manifold M,
 Open subset U of the 2-dimensional topological manifold M,
 Subset V of the 2-dimensional topological manifold M]

The method subsets() returns a set instead of a list:

sage: M.subsets()  # random (set output)
{Subset V of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M,
 Open subset U of the 2-dimensional topological manifold M}

manifold()

Return the ambient manifold of self.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: A = M.subset('A')
sage: A.manifold()
2-dimensional topological manifold M
sage: A.manifold() is M
True
sage: B = A.subset('B')
sage: B.manifold() is M
True

An alias is ambient:

sage: A.ambient() is A.manifold()
True

open_covers()

Return the list of open covers of the current subset.

If the current subset, \(A\) say, is a subset of the manifold \(M\), an open cover of \(A\) is list (indexed set) \((U_i)_{i\in I}\) of open subsets of \(M\) such that

\[A \subset \bigcup_{i \in I} U_i.\]

If \(A\) is open, we ask that the above inclusion is actually an identity:

\[A = \bigcup_{i \in I} U_i.\]

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: M.open_covers()
[[2-dimensional topological manifold M]]
sage: U = M.open_subset('U')
sage: U.open_covers()
[[Open subset U of the 2-dimensional topological manifold M]]
sage: A = U.open_subset('A')
sage: B = U.open_subset('B')
sage: U.declare_union(A,B)
sage: U.open_covers()
[[Open subset U of the 2-dimensional topological manifold M],
 [Open subset A of the 2-dimensional topological manifold M,
  Open subset B of the 2-dimensional topological manifold M]]
sage: V = M.open_subset('V')
sage: M.declare_union(U,V)
sage: M.open_covers()
[[2-dimensional topological manifold M],
 [Open subset U of the 2-dimensional topological manifold M,
  Open subset V of the 2-dimensional topological manifold M],
 [Open subset A of the 2-dimensional topological manifold M,
  Open subset B of the 2-dimensional topological manifold M,
  Open subset V of the 2-dimensional topological manifold M]]

point(coords=None, chart=None, name=None, latex_name=None)

Define a point in self.

See ManifoldPoint for a complete documentation.

INPUT:

• coords – the point coordinates (as a tuple or a list) in the chart specified by chart

• chart – (default: None) chart in which the point coordinates are given; if None, the coordinates are assumed to refer to the default chart of the current subset

• name – (default: None) name given to the point

• latex_name – (default: None) LaTeX symbol to denote the point; if None, the LaTeX symbol is set to name

OUTPUT:

• the declared point, as an instance of ManifoldPoint

EXAMPLES:

Points on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: c_xy.<x,y> = M.chart()
sage: p = M.point((1,2), name='p'); p
Point p on the 2-dimensional topological manifold M
sage: p in M
True
sage: a = M.open_subset('A')
sage: c_uv.<u,v> = a.chart()
sage: q = a.point((-1,0), name='q'); q
Point q on the 2-dimensional topological manifold M
sage: q in a
True
sage: p._coordinates
{Chart (M, (x, y)): (1, 2)}
sage: q._coordinates
{Chart (A, (u, v)): (-1, 0)}

retract(p)

Return the retract of p to self.

INPUT:

• p – point of the ambient manifold

OUTPUT:

• the same point, considered as a point of the subset

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: A = M.open_subset('A', coord_def={X: x>0})
sage: p = M((1, -2)); p
Point on the 2-dimensional topological manifold M
sage: p.parent()
2-dimensional topological manifold M
sage: q = A.retract(p); q
Point on the 2-dimensional topological manifold M
sage: q.parent()
Open subset A of the 2-dimensional topological manifold M
sage: q.coord()
(1, -2)
sage: (q == p) and (p == q)
True

Of course, if the point does not belong to A, the retract method fails:

sage: p = M((-1, 3))  #  x < 0, so that p is not in A
sage: q = A.retract(p)
Traceback (most recent call last):
...
ValueError: the Point on the 2-dimensional topological manifold M
 is not in Open subset A of the 2-dimensional topological manifold M

subset(name, latex_name=None, is_open=False)

Create a subset of the current subset.

INPUT:

• name – name given to the subset

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

• is_open – (default: False) if True, the created subset is assumed to be open with respect to the manifold’s topology

OUTPUT:

• the subset, as an instance of ManifoldSubset, or of the derived class TopologicalManifold if is_open is True

EXAMPLES:

Creating a subset of a manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: a = M.subset('A'); a
Subset A of the 2-dimensional topological manifold M

Creating a subset of A:

sage: b = a.subset('B', latex_name=r'\mathcal{B}'); b
Subset B of the 2-dimensional topological manifold M
sage: latex(b)
\mathcal{B}

We have then:

sage: b.is_subset(a)
True
sage: b in a.subsets()
True

subsets()

Return the set of subsets that have been defined on the current subset.

OUTPUT:

• a Python set containing all the subsets that have been defined on the current subset

Note

To get the subsets as a list, used the method list_of_subsets() instead.

EXAMPLES:

Subsets of a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: U = M.open_subset('U')
sage: V = M.subset('V')
sage: M.subsets()  # random (set output)
{Subset V of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M,
 Open subset U of the 2-dimensional topological manifold M}
sage: type(M.subsets())
<... 'frozenset'>
sage: U in M.subsets()
True

The method list_of_subsets() returns a list (sorted alphabetically by the subset names) instead of a set:

sage: M.list_of_subsets()
[2-dimensional topological manifold M,
 Open subset U of the 2-dimensional topological manifold M,
 Subset V of the 2-dimensional topological manifold M]

superset(name, latex_name=None, is_open=False)

Create a superset of the current subset.

A superset is a manifold subset in which the current subset is included.

INPUT:

• name – name given to the superset

• latex_name – (default: None) LaTeX symbol to denote the superset; if none are provided, it is set to name

• is_open – (default: False) if True, the created subset is assumed to be open with respect to the manifold’s topology

OUTPUT:

• the superset, as an instance of ManifoldSubset or of the derived class TopologicalManifold if is_open is True

EXAMPLES:

Creating some superset of a given subset:

sage: M = Manifold(2, 'M', structure='topological')
sage: a = M.subset('A')
sage: b = a.superset('B'); b
Subset B of the 2-dimensional topological manifold M
sage: b.list_of_subsets()
[Subset A of the 2-dimensional topological manifold M,
 Subset B of the 2-dimensional topological manifold M]
sage: a._supersets # random (set output)
{Subset B of the 2-dimensional topological manifold M,
 Subset A of the 2-dimensional topological manifold M,
 2-dimensional topological manifold M}

The superset of the whole manifold is itself:

sage: M.superset('SM') is M
True

Two supersets of a given subset are a priori different:

sage: c = a.superset('C')
sage: c == b
False

union(other, name=None, latex_name=None)

Return the union of the current subset with another subset.

INPUT:

• other – another subset of the same manifold

• name – (default: None) name given to the union in the case the latter has to be created; the default is self._name union other._name

• latex_name – (default: None) LaTeX symbol to denote the union in the case the latter has to be created; the default is built upon the symbol \(\cup\)

OUTPUT:

• instance of ManifoldSubset representing the subset that is the union of the current subset with other

EXAMPLES:

Union of two subsets:

sage: M = Manifold(2, 'M', structure='topological')
sage: a = M.subset('A')
sage: b = M.subset('B')
sage: c = a.union(b); c
Subset A_union_B of the 2-dimensional topological manifold M
sage: a._supersets  # random (set output)
set([subset 'A_union_B' of the 2-dimensional manifold 'M',
     2-dimensional manifold 'M',
     subset 'A' of the 2-dimensional manifold 'M'])
sage: b._supersets  # random (set output)
set([subset 'B' of the 2-dimensional manifold 'M',
     2-dimensional manifold 'M',
     subset 'A_union_B' of the 2-dimensional manifold 'M'])
sage: c._subsets  # random (set output)
set([subset 'A_union_B' of the 2-dimensional manifold 'M',
    subset 'A' of the 2-dimensional manifold 'M',
    subset 'B' of the 2-dimensional manifold 'M'])

Some checks:

sage: a.is_subset(a.union(b))
True
sage: b.is_subset(a.union(b))
True
sage: a.union(b) is b.union(a)
True
sage: a.union(a.union(b)) is a.union(b)
True
sage: (a.union(b)).union(a) is a.union(b)
True
sage: a.union(M) is M
True
sage: M.union(a) is M
True