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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 main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. [9] In particular this applies to any action of a finite group ; thus a manifold with boundary carries a natural orbifold structure, since it is the ...
By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity. That is, a group G is a locally compact space if and only if the identity element has a compact neighborhood. It follows that there is a local base of compact neighborhoods at every point.
An informal definition is commonly used in physics textbooks and introductory lectures. It defines a supermanifold as a manifold with both bosonic and fermionic coordinates. . Locally, it is composed of coordinate charts that make it look like a "flat", "Euclidean" supersp
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]
Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry.
The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors.
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 ...