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Encyclopedia > Connectedness

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component). Look up connection, connected, connectivity in Wiktionary, the free dictionary. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

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Connectedness in topology

Main article: Connected space

A topological space is said to be connected if it cannot be contained in two disjoint nonempty open sets. 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. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...


Other notions of connectedness

Fields of mathematics are typically concerned with special kinds of objects. Often such an object is said to be connected if, when it is considered as a topological space, it is a connected space. Thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces, and their components are the topological components. Sometimes it is convenient to restate the definition of connectedness in such fields. For example, a graph is said to be connected if each pair of vertices in the graph is joined by a path. This definition is equivalent to the topological one, as applied to graphs, but it is easier to deal with in the context of graph theory. Graph theory also offers a context-free measure of connectedness, called the clustering coefficient. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics and computer science, graph theory studies the properties of graphs. ... This article just presents the basic definitions. ... In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. ... A drawing of a graph. ... Example clustering coefficient on an undirected graph for the shaded node i. ...


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. For example, in category theory, a category is said to be connected if each pair of objects in it is joined by a morphism. Thus, a category is connected if it is, intuitively, all one piece. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...


There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space connected if each pair of points in it is joined by a path. However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used; spaces with this property are said to be path connected. In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...


Terms involving connected are also used for properties that are related to, but clearly different from, connectedness. For example, a path-connected topological space is simply connected if each loop (path from a point to itself) in it is contractible; that is, intuitively, if there is essentially only one way to get from any point to any other point. Thus, a sphere and a disk are each simply connected, while a torus is not. As another example, a directed graph is strongly connected if each ordered pair of vertices is joined by a directed path (that is, one that "follows the arrows"). In topology, a geometrical object or space is called simply connected if it is path-connected and every path between two points can be continuously transformed into every other. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... For other uses, see Sphere (disambiguation). ... In geometry, a disk is the region in a plane contained inside of a circle. ... In geometry, a torus (pl. ... This article just presents the basic definitions. ... Graph with SCC marked A directed graph is called strongly connected if for every pair of vertices u and v there is a path from u to v and a path from v to u. ... In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ...


Other concepts express the way in which an object is not connected. For example, a topological space is totally disconnected if each of its components is a single point. In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...


Connectivity

Properties and parameters based on the idea of connectedness often involve the word connectivity. For example, in graph theory, a connected graph is one from which we must remove at least one vertex to create a disconnected graph. In recognition of this, such graphs are also said to be 1-connected. Similarly, a graph is 2-connected if we must remove at least two vertices from it, to create a disconnected graph. A 3-connected graph requires the removal of at least three vertices, and so on. The connectivity of a graph is the minimum number of vertices that must be removed, to disconnect it. Equivalently, the connectivity of a graph is the greatest integer k for which the graph is k-connected. In mathematics and computer science the connectivity of graphs is one of the basic concepts of graph theory. ... A drawing of a graph. ... In mathematics and computer science, graph theory studies the properties of graphs. ... In mathematics and computer science the connectivity of graphs is one of the basic concepts of graph theory. ...


While terminology varies, noun forms of connectedness-related properties often include the term connectivity. Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple connectedness. On the other hand, in fields without a formally defined notion of connectivity, the word may be used as a synonym for connectedness. In linguistics, a noun or noun substantive is a lexical category which is defined in terms of how its members combine with other grammatical kinds of expressions. ...


Another example of connectivity can be found in regular tilings. Here, the connectivity describes the number of neighbors accessible from a single tile: Mission, or barrel, roof tiles A tile is a manufactured piece of hard-wearing material such as ceramic, stone, porcelain, metal or even glass. ...

See also


  Results from FactBites:
 
Connectedness - Wikipedia, the free encyclopedia (765 words)
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".
However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold.
For example, in graph theory, a connected graph is one from which we must remove at least one vertex to create a disconnected graph.
  More results at FactBites »

 
 

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