N-connected space

:"n-connected redirects here; for the concept in graph theory see Connectivity (graph theory)."

In the mathematical branch of topology, a topological space "X" is said to be "n"-connected if and only if it is path-connected and its first "n" homotopy groups vanish identically, that is

:$pi\_i(X)\; equiv\; 0~,\; quad\; 1leq\; ileq\; n\; ,$

where the left-hand side denotes the "i"-th homotopy group. The requirement of being path-connected can also be expressed as "0-connectedness", when defining the "0th homotopy group"

:$pi\_0(X)\; :=\; [S^0,\; X]\; .$

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 "x₁" and "x₂" in "X" can be connected with a continuous path which starts in "x₁" and ends in "x₂", which is equivalent to the assertion that every mapping from "S⁰" (a discrete set of two points) to "X" can be deformed continuously to a constant map. With this definition, we can define "X" to be n-connected if and only if

:$pi\_i(X)\; equiv\; 0,\; quad\; 0leq\; ileq\; n.$

Examples and applications

* As described above, a space "X" is "0-connected" if and only if it is path-connected.
* A space is "1-connected" if and only if it is simply connected. Thus, the term "n-connected" is a natural generalization of being path-connected or simply connected.

It is obvious from the definition that an "n"-connected space "X" is also "i"-connected for all "iThe concept of "n"-connectedness is used in the Hurewicz theorem which describes the relation between singular homology and the higher homotopy groups.

ee also

* connected space
* simply connected
* path-connected
* homotopy group