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)

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

EXAMPLES:

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


The closure of the subset $$D$$ is a subset of every closed superset of $$D$$:

sage: S = D.superset('S')
sage: S.declare_closed()
sage: 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: 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