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In mathematics, connectedness [1] 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 ...
A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater. More precisely, any graph G (complete or not) is said to be k -vertex-connected if it contains at least k + 1 vertices, but does not contain a set of k − 1 vertices whose removal disconnects the graph; and κ ( G ) is defined as the largest k such ...
A path-connected space is a stronger notion of connectedness, requiring the structure of a path. A path from a point to a point in a topological ...
A space is totally disconnected if it has no connected subset with more than one point. Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected. Locally path-connected.
In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex-connectivity , or just connectivity , of a graph is the largest k for which the graph is k -vertex-connected.
In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected. A sphere is simply connected because every loop can be contracted (on the surface) to a point. The definition rules out only handle-shaped holes. A sphere (or ...
First, a word about cause versus risk. On a cellular level, alcohol is carcinogenic due to the ways it damages cells. When it comes to a whole person, alcohol is one of many factors — which also ...
An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field. A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle.