Now, I may have only slept two hours last night and would currently struggle to discern a 'proof' by induction of FLT from a piece of genuine mathematics, but that doesn't stop mathematics from bugging me. At present I am puzzled by something I saw on MO this morning...
The linked question concerns the Hilbert cube $[0,1]^\mathbb{N}$ (an infinite product of intervals) and the existence of space filling curves thereof- that is: continuous images of the unit circle that are surjections on the Hilbert cube. The accepted answer, together with another answer (which actually constructs such a map) and various comments, seems to allude toward an answer in the affirmative. However, the linked theorem (the 'Hahn–Mazurkiewicz theorem') which states:
A nonempty Hausdorff topological space is a continuous image of the unit interval if and only if it is a compact, connected, locally connected second-countable space.
seems in direct contradiction to this since (and I may be mistaken for reasons explained above):
- The Hilbert cube is a subset of a normed space and hence a metric space
- The sequence $(1,0,0...), (0,1,0...), (0,0,1,...)$ has no convergent subsequence
- So the Hilbert cube is not sequentially compact, therefore non-compact (the two are equivalent in metric spaces).
Which seems at odds with the only if portion of the theorem's statement. Maybe this is wikipedia taking me for a ride. Maybe I am just hallucinating a portion of this argument. Either way, this is annoying me. Thanks in advance for clearing this up...