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Examples of such spaces include the Cantor space and Baire space. Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers where = {,} is given the discrete topology. Such a space is sometimes called a Cantor cube.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine , Euclidean , absolute , and hyperbolic geometry (but not for projective geometry).
A general paradigm in group theory is that a group G should be studied via its group representations.A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M.
Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring with any topological space. Every continuous map: determines a homomorphism from the cohomology ring of to that of ; this puts strong restrictions on the possible maps from to .
In the zeroth-order example above, the quantity "a few" was given, but in the first-order example, the number "4" is given. A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example:
For the corresponding definition in terms of spheres, define the sum + of maps ,: to be composed with h, where is the map from to the wedge sum of two n-spheres that collapses the equator and h is the map from the wedge sum of two n-spheres to X that is defined to be f on the first sphere and g on the second.
In algebraic geometry and complex analytic geometry, coherent sheaves are a class of sheaves of particular geometric importance. For example, an algebraic vector bundle (on a locally Noetherian scheme ) or a holomorphic vector bundle (on a complex analytic space ) can be viewed as a coherent sheaf, but coherent sheaves have the advantage over ...
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