In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.

An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial.

Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".

All definitions below consider a topological space X.

A hole in X is, informally, a thing that prevents some suitably placed sphere from continuously shrinking to a point. Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,

In general, for every integer d, ${\displaystyle {\text{conn}}_{\pi }(S^{d})=d-1}$ (and ${\displaystyle \eta _{\pi }(S^{d})=d+1}$) The proof requires two directions:

A space X is called n-connected, for n ≥ 0, if it is non-empty, and all its homotopy groups of order d ≤ n are the trivial group: $${\displaystyle \pi _{d}(X)\cong 0,\quad -1\leq d\leq n,}$$ where ${\displaystyle \pi _{i}(X)}$ denotes the i-th homotopy group and 0 denotes the trivial group. The two definitions are equivalent. The requirement for an n-connected space consists of requirements for all d ≤ n:

The requirements of being non-empty and path-connected can be interpreted as (−1)-connected and 0-connected, respectively, which is useful in defining 0-connected and 1-connected maps, as below. The 0th homotopy set can be defined as: