Topological Closures of Manifold Subsets - Manifolds

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# 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