[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube"
[0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] × …
A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …
We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.
—Jean-Marc Cordier and Timothy Porter, Shape Theory
(I rearranged their words liberally but the substance is theirs.)
in R
do: prod( factorial( 1/ 1:10e4) )
to see the volume of Hilbert’s cube → 0.
(Source: amzn.to)