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Graph theory also offers a context-free measure of connectedness, called the clustering coefficient. Other fields of mathematics are concerned with objects that are rarely considered as topological spaces. Nonetheless, definitions of connectedness often reflect the topological meaning in some way.
As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets ( Muscat & Buhagiar 2006 ).
Connectedness is preserved by graph homomorphisms. If G is connected then its line graph L(G) is also connected. A graph G is 2-edge-connected if and only if it has an orientation that is strongly connected. Balinski's theorem states that the polytopal graph (1-skeleton) of a k-dimensional convex polytope is a k-vertex-connected graph. [13]
Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space.
Social Connectedness Scale [49] This scale was designed to measure general feelings of social connectedness as an essential component of belongingness. Items on the Social Connectedness Scale reflect feelings of emotional distance between the self and others, and higher scores reflect more social connectedness.
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 ...
Connectedness is a basic measure in many areas of mathematical science and social sciences. In graph theory, two vertices are said to be connected if there is a path between them. In topology , two points are connected if there is a continuous function that could move from one point to another continuously.
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected [1]) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question.