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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.
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.
Having established uniform convergence on compact sets, the harmonicity of the limit is an immediate corollary of the fact that the mean value property (automatically preserved by uniform convergence) fully characterizes harmonic functions among continuous functions.
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.
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.
In mathematics, the connectedness theorem may be one of Deligne's connectedness theorem; Fulton–Hansen connectedness theorem; Grothendieck's connectedness theorem;
They are called the strong law of large numbers and the weak law of large numbers. [ 16 ] [ 1 ] Stated for the case where X 1 , X 2 , ... is an infinite sequence of independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E( X 1 ) = E( X 2 ) = ... = μ , both versions of the law state that the ...
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.