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Connected Mathematics is a comprehensive mathematics program intended for U.S. students in grades 6–8.The curriculum design, text materials for students, and supporting resources for teachers were created and have been progressively refined by the Connected Mathematics Project (CMP) at Michigan State University with advice and contributions from many mathematics teachers, curriculum ...
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 ...
It is locally connected if it has a base of connected sets. It can be shown that a space is locally connected if and only if every component of every open set of is open. Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if ...
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 ...
This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen. Now let X {\displaystyle X} be an infinite set under the discrete metric – that is, two points p , q ∈ X {\displaystyle p,q\in X} have distance 1 if they're not the same point, and 0 ...
A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group). n -connectivity for spaces can in turn be defined in terms of n -connectivity of maps: a space X with basepoint x 0 is an n -connected space if and only if the inclusion of the ...
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Connected component may refer to: Connected component (graph theory) , a set of vertices in a graph that are linked to each other by paths Connected component (topology) , a maximal subset of a topological space that cannot be covered by the union of two disjoint non-empty open sets