nLab 0-sphere




The 0-sphere S 0S^0 is the n-sphere for n=0n = 0: The disjoint union of two points,

S 0**. S^0 \;\;\simeq\;\; \ast \sqcup \ast \,.

This definition (even if possibly not be familiar from standard texts of point-set topology) is certainly justified by the fact that the 2-element set is clearly:

In fact, with suspension SX\mathrm{S}X understood as the homotopy pushout of the terminal map X*X \to \ast along itself, one has also

S 0SS 1 S^0 \;\coloneqq\; \mathrm{S} S^{-1}

for S 1S^{-1} \,\coloneqq\, \varnothing the empty set.

These spheres of degenerate dimensions play an imprtant role in homotopy theory, see for instance the generating cofibrations in the classical model structure on topological spaces (here).

Of course, the underlying object of S 0S^0, simple as it is, plays various other roles, too:

Last revised on January 5, 2023 at 21:52:51. See the history of this page for a list of all contributions to it.