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In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant. A uniform space U is called uniformly disconnected if it is not uniformly connected.
A topological space X is path-connected if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points x 1 and x 2 in X can be connected with a continuous path which starts in x 1 and ends in x 2, which is equivalent to the assertion that every mapping from S 0 (a discrete set of two points) to X ...
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1.
A topological space is said to be connected if it is not the union of two disjoint nonempty open sets. [2] A set is open if it contains no point lying on its boundary; thus, in an informal, intuitive sense, the fact that a space can be partitioned into disjoint open sets suggests that the boundary between the two sets is not part of the space, and thus splits it into two separate pieces.
Given some point in a topological space , the union of any collection of connected subsets such that each contains will once again be a connected subset. The connected component of a point in is the union of all connected subsets of that contain ; it is the unique largest (with respect to ) connected subset of that contains .
The idea of composition in the fundamental group is that of traveling the first path and the second in succession, or, equivalently, setting their two domains together. The concept of composition that we want for the n-th homotopy group is the same, except that now the domains that we stick together are cubes, and we must glue them along a face.
In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups , but the concept is designed to formulate the weakest axioms needed for most proofs ...
Zariski's connectedness theorem, a generalization of Zariski's main theorem Topics referred to by the same term This disambiguation page lists mathematics articles associated with the same title.
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