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## Continuous bijective maps with connected proper domain and compact range (all spaces are metric) [closed]

Hie everyone, The question is simple, but I haven't found the solution yet... Is it true that a continuous bijective map with connected proper (i.e. with compact balls) domain and compact range is a homeomorphism (all topological spaces are assumed to be metric spaces)? Can we replace connected by second countable? What if we suppose all the spaces are length/geodesic spaces?

 Of course not, just take a line with irrational slope in the flat torus $S^1\times S^1$, sorry! – Aurelien Mar 16 2011 at 12:50 This counter example is false since the map is not surjective. The counter example I was looking for is the map: $$f:[0,1]\rightarrow\mathcal{S}_1;\ x\mapsto e^{2i\pi}$$ – Aurelien Mar 25 2011 at 13:48 Ok, this one is not injective now... – Aurelien Mar 25 2011 at 13:50