For all n, j > 0: (Bj is n-connected.
Indeed any convex set (including ∅) is n-connected.)
For all n, j > 0: Sj is n-connected just if n ≠ j+1.
“path connected” is 1-connected.
simply connected is (1-connected and 2-connected)
Some sets and their connectivity: The nth entry on the right (n>0) is whether the body described below is n-connected:
|S0||F T T T …|
|S1||T F T T …|
|S2||T T F T …|
|Bn||T T T T …|
|T2: Torus||T F T T …|
|ST2 = 3D volume bounded|
by T2: Torus
|T F T T …|
|Q = ST2 omitting a|
small solid void:
|T F F T …|
|Two disjoint copies of B2||F T T T …|
|Two disjoint copies of ST2||F F T T …|
|Two disjoint copies of S2||F T F T …|
|Two disjoint copies of Q.||F F F T …|
If sets P' and Q' are base families (they each serve as a base for topologies P and Q respectively) then P'×Q' is a base family which serves as a base for R the product topology for P and Q. I think that for each positive integer n, R is n-connected just if both P' and Q' are both n-connected. I think this goes for infinite products too.
It seems the definition goes over smoothly to ω-connected by reasoning about Hilbert space over the reals and the unit ball there. For Hilbert space we must use balls or simplexes, not cubes.