Julian answered off-list: On the Klein bottle? No. Any continuous bijection of compact Hausdorff spaces has a continuous inverse. More generally? Sure. Take 0->0, n->1/n for n≥1 from the discrete space N to R; it's a continuous bijection onto its image, but the inverse is not continuous. Or take t->exp(2πit), from [0,1) to the unit circle. Julian On Thu, Dec 20, 2018 at 10:51 PM Bill Gosper <billgosper@gmail.com> wrote:

DanA: Consider the space Homeo(K) of self-homeomorphisms of the Klein bottle. I.e., Homeo(K) consists of all continuous bijections

h : K —> K

having a continuous inverse. __________ Umm, can somebody show me a continuous bijection with a discontinuous inverse? —rwg