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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 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.
Thus is a sharp threshold for the connectedness of G(n, p). Further properties of the graph can be described almost precisely as n tends to infinity. For example, there is a k ( n ) (approximately equal to 2log 2 ( n )) such that the largest clique in G ( n , 0.5) has almost surely either size k ( n ) or k ( n ) + 1.
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 .
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.
Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order.
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence.