User blog comment:Edwin Shade/Higher-Dimensional Spaces/@comment-28606698-20171016200813

1. The sphere which loops 2D-space on itself is 2D-sphere (not 3D) or in other words 2-dimensional manifold that can be embedded in Euclidean 3-space (similarly as a circle is 1-sphere)

2. Your definition written for cases when amount of dimensions is a successor ordinal (1,2,3, or omega+1), how about a limit ordinal, for example omega?