### New definition for connected

For 1 ≤ n the n-ball is Bn = {(x1, … xn) | Σ[1≤j≤n]xj2 ≤ 1}. Said otherwise the n-ball is the (closed) set of points in an n-dimensional Euclidean space within distance 1 of the origin. I propose here a definition of n-connected that is different from this.
For n > 0 a topological space X is n-connected just if every map from the boundary of the n-ball into X can be extended to a map from the ball into X.
1-connected is the conventional ‘path connected’. A set that is 1-connected and 2-connected is simply connected. Where Cn is the conventional n-connectedness, Cn x iff for all k<n x is n-connected. My notion is more discriminating.

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 boundedby T2: Torus T F T T … Q = ST2 omitting asmall 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 …
For any base family B, let T(B) = {b | b⊆B} which are the open sets of a topology.

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.

I think that all connectedness combinations are possible within Rn for some n except that if n > m then any set in Rm is n-connected.

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.