Continuous bijections and their inverses












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Are there any theorems characterizing the continuity of the inverses of bijections between certain classes of particular spaces. E.g. it seems to me the inverse of any continuous bijection between connected subsets of $mathbb{R}$ is continouos. It seems that most counterexamples require highly specific spaces. I'm thinking there might be something along the lines of any continuous bijection between homeomorphic spaces has a continuous inverse. Any of this make sense?






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    Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
    $endgroup$
    – freakish
    Dec 27 '18 at 18:28












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    I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:Xto X$ whose inverse is not continuous.
    $endgroup$
    – DanielWainfleet
    Jan 3 at 10:26


















1












$begingroup$


Are there any theorems characterizing the continuity of the inverses of bijections between certain classes of particular spaces. E.g. it seems to me the inverse of any continuous bijection between connected subsets of $mathbb{R}$ is continouos. It seems that most counterexamples require highly specific spaces. I'm thinking there might be something along the lines of any continuous bijection between homeomorphic spaces has a continuous inverse. Any of this make sense?






general-topology







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
    $endgroup$
    – freakish
    Dec 27 '18 at 18:28












  • $begingroup$
    I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:Xto X$ whose inverse is not continuous.
    $endgroup$
    – DanielWainfleet
    Jan 3 at 10:26
















1












1








1





$begingroup$


Are there any theorems characterizing the continuity of the inverses of bijections between certain classes of particular spaces. E.g. it seems to me the inverse of any continuous bijection between connected subsets of $mathbb{R}$ is continouos. It seems that most counterexamples require highly specific spaces. I'm thinking there might be something along the lines of any continuous bijection between homeomorphic spaces has a continuous inverse. Any of this make sense?






general-topology







share|cite|improve this question









$endgroup$




Are there any theorems characterizing the continuity of the inverses of bijections between certain classes of particular spaces. E.g. it seems to me the inverse of any continuous bijection between connected subsets of $mathbb{R}$ is continouos. It seems that most counterexamples require highly specific spaces. I'm thinking there might be something along the lines of any continuous bijection between homeomorphic spaces has a continuous inverse. Any of this make sense?






general-topology




general-topology






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asked Dec 27 '18 at 16:36









Keefer RowanKeefer Rowan

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20510








  • 1




    $begingroup$
    Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
    $endgroup$
    – freakish
    Dec 27 '18 at 18:28












  • $begingroup$
    I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:Xto X$ whose inverse is not continuous.
    $endgroup$
    – DanielWainfleet
    Jan 3 at 10:26
















  • 1




    $begingroup$
    Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
    $endgroup$
    – freakish
    Dec 27 '18 at 18:28












  • $begingroup$
    I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:Xto X$ whose inverse is not continuous.
    $endgroup$
    – DanielWainfleet
    Jan 3 at 10:26










1




1




$begingroup$
Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
$endgroup$
– freakish
Dec 27 '18 at 18:28






$begingroup$
Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
$endgroup$
– freakish
Dec 27 '18 at 18:28














$begingroup$
I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:Xto X$ whose inverse is not continuous.
$endgroup$
– DanielWainfleet
Jan 3 at 10:26






$begingroup$
I can give you an example of a complex normed linear space $X$ and a continuous linear bijection $f:Xto X$ whose inverse is not continuous.
$endgroup$
– DanielWainfleet
Jan 3 at 10:26












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Indeed, in particular cases the inverse of a continuous bijection $f:Xto Y$ is continuous. Namely, when $X$ is a compact, $Y$ is Hausdorff (well-known and very easy to prove) or when $X$ is an open subset of $Bbb R^n$ and $YsubsetBbb R^n$, by invariance of domain.






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    $begingroup$

    Indeed, in particular cases the inverse of a continuous bijection $f:Xto Y$ is continuous. Namely, when $X$ is a compact, $Y$ is Hausdorff (well-known and very easy to prove) or when $X$ is an open subset of $Bbb R^n$ and $YsubsetBbb R^n$, by invariance of domain.






    share|cite|improve this answer









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      1












      $begingroup$

      Indeed, in particular cases the inverse of a continuous bijection $f:Xto Y$ is continuous. Namely, when $X$ is a compact, $Y$ is Hausdorff (well-known and very easy to prove) or when $X$ is an open subset of $Bbb R^n$ and $YsubsetBbb R^n$, by invariance of domain.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Indeed, in particular cases the inverse of a continuous bijection $f:Xto Y$ is continuous. Namely, when $X$ is a compact, $Y$ is Hausdorff (well-known and very easy to prove) or when $X$ is an open subset of $Bbb R^n$ and $YsubsetBbb R^n$, by invariance of domain.






        share|cite|improve this answer









        $endgroup$



        Indeed, in particular cases the inverse of a continuous bijection $f:Xto Y$ is continuous. Namely, when $X$ is a compact, $Y$ is Hausdorff (well-known and very easy to prove) or when $X$ is an open subset of $Bbb R^n$ and $YsubsetBbb R^n$, by invariance of domain.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 3 at 4:34









        Alex RavskyAlex Ravsky

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