nLab n-connected space

-(simply) connected spaces


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


nn-(simply) connected spaces


An nn-connected space is a generalisation of the pattern:

For the general concept see at n-connected object of an (infinity,1)-topos.


A topological space XX is nn-connected or nn-simply connected if its homotopy groups are trivial up to degree nn.

More explicitly, XX is precisely kk-connected if every continuous map to XX from the kk-sphere extends to a continuous map to XX from the k+1k+1-disk. Then XX is nn-(simply) connected if XX is precisely kk-connected for 1kn-1 \leq k \leq n.

Special cases

  • Any space is (2)(-2)-simply connected.

  • A space is (1)(-1)-simply connected precisely if it has an element; that is if it is inhabited.

  • A space is 00-simply connected precisely if it is path-connected.

  • A space is 11-simply connected precisely if it is simply connected.

  • A space is \infty-simply connected precisely if it is weakly contractible.


The traditional terminology is ‘nn-connected’, but this violates the rule that ‘11-foo’ should mean the same as ‘foo’. This can be fixed by saying ‘nn-simply connected’ instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.


An nn-connected topological space is precisely an n-connected object in the (∞,1)-topos ∞Gpd, presented by the model category Top of topological spaces.

Last revised on October 28, 2019 at 22:18:31. See the history of this page for a list of all contributions to it.