n-connected space
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- 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
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"
- π0(X): = [S0,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
[edit] 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 i<n.
The concept of n-connectedness is used in the Hurewicz theorem which describes the relation between singular homology and the higher homotopy groups.
