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If () = for all but a finite number of points , then is said to have finite support. If the set X {\displaystyle X} has an additional structure (for example, a topology ), then the support of f {\displaystyle f} is defined in an analogous way as the smallest subset of X {\displaystyle X} of an appropriate type such that f {\displaystyle f ...
A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points ,, …, if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to .
The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straightedge construction (an endpoint of a line segment or crossing point of two ...
Equinumerosity is compatible with the basic set operations in a way that allows the definition of cardinal arithmetic. [1] Specifically, equinumerosity is compatible with disjoint unions: Given four sets A, B, C and D with A and C on the one hand and B and D on the other hand pairwise disjoint and with A ~ B and C ~ D then A ∪ C ~ B ∪ D.
The action of G on X is called transitive if for any two points x, y ∈ X there exists a g ∈ G so that g ⋅ x = y. The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given x, y ∈ X the element g in the definition of transitivity is unique.
A diagram illustrating the great-circle distance (in cyan) and the straight-line distance (in red) between two points P and Q on a sphere. To see the utility of different notions of distance, consider the surface of the Earth as a set of points.
For example, the closed subscheme x 3 + y 3 = z 3 of is an elliptic curve over the rational numbers. The line with two origins (over a field k) is the scheme defined by starting with two copies of the affine line over k, and gluing together the two open subsets A 1 − 0 by the identity map. This is a simple example of a non-separated scheme.
Hasse diagram of the natural numbers, partially ordered by "x≤y if x divides y".The numbers 4 and 6 are incomparable, since neither divides the other. In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true.