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For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex. The underlying space, sometimes called the carrier of a simplicial complex, is the union of
The underlying space of a fan is the union of its cones and is denoted by | |. The toric variety of a fan of strongly convex rational cones is given by taking the affine toric varieties of its cones and gluing them together by identifying U σ {\displaystyle U_{\sigma }} with an open subvariety of U τ {\displaystyle U_{\tau }} whenever σ ...
If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is /, where is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M.
Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S 1 is globally isomorphic to S 1 × R, since there is a global nonzero vector ...
In Bourbaki's terms, [2] "topological space" is an underlying structure of the "Euclidean space" structure. Similar ideas occur in category theory : the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category.
The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...
Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. [1]
An orbifold path is a path in the underlying space provided with an explicit piecewise lift of path segments to orbifold charts and explicit group elements identifying paths in overlapping charts; if the underlying path is a loop, it is called an orbifold loop. Two orbifold paths are identified if they are related through multiplication by ...