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> there exists no continuous epimorphism f: R^n -> R^m, where m > n Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve https://en.wikiped
by waqf 9y ago
> there exists no continuous epimorphism f: R^n -> R^m, where m > n
Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve https://en.wikipedia.org/wiki/Space-filling_curve?
- dahart 9y agoRight from that article you linked: "A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut-point, and so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points."
- waqf 9y agoYeah, a non-self-intersecting map cannot, but OP didn't specify that, only "epimorphic" which certainly can. Moreover OP's argument specifically proves too much, because space-filling curves (as described in the article) have a range with positive Borel measure.