User blog comment:Edwin Shade/Higher-Dimensional Spaces/@comment-1605058-20171016210537

While your definitions are not precise enough to really make mathematical claims about them, I feel like with what you have, the \(\omega+1\) space is going to be the same as \(\omega\) space - I can't really imagine there being any intrinsic difference between a space determined by \(\omega+1\) coordinates \([b_1,b_2,\dots]\cup[a_1]\) and a space determined by \(\omega\) (simply reordered) coordinates \([a_1,b_1,b_2,\dots]\). These two spaces should have the same geometric properties (perhaps formalized by using a ). In particular, if you somehow include an "\(\omega\) space that loop back on itself" in \(\omega+1\) space, then, reordering coordinates, you should be able to include such a space in the \(\omega\) space itself.