Applications
The concept of n-connectedness is used in the Hurewicz theorem which describes the relation between singular homology and the higher homotopy groups.
In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersions into a more general topological space, such as the space of all continuous maps between two associated spaces are n-connected are said to satisfy a homotopy principle or "h-principle". There are a number of powerful general techniques for proving h-principles.
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