N connected space and n connected map

A (−1)-connected space is a nonempty space. For n≥0, an n-connected space is nonempty and path-connected and has ${\pi }_i(X,x)=0$ for every $x\in X$ and $1\le i\le n$. Nonemptiness is additional to Paths, path-connected spaces and path components, whose path-connectedness condition is vacuous on the empty space.

A (−1)-connected map imposes no condition. A continuous map $f:X\to Y$ of CGWH spaces is 0-connected if ${\pi }_0(X)\to {\pi }_0(Y)$ is surjective. For n≥1 it is n-connected if this component-surjectivity condition holds and ${\pi }_i(M_f,j(X),j(x))$ is the distinguished singleton for every $x\in X$ and $1\le i\le n$. The cylinder factorization is Mapping cylinder factorization, and the relative groups and pointed sets are those of Long exact sequence of relative homotopy groups. In degree one, triviality means a singleton pointed set. The relative condition is imposed at every source basepoint $x\in X$. The separate component-surjectivity condition ensures that no target component outside the image is omitted.