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:
sage.manifolds.subset.ManifoldSubset
Topological closure of a manifold subset in the topology of the manifold.
INPUT:
subset
– aManifoldSubset
name
– (default: computed from the name of the subset) string; name (symbol) given to the closurelatex_name
– (default:None
) string; LaTeX symbol to denote the subset; if none is provided, it is set toname
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