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?
general-topology
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
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add a comment |
$begingroup$
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
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
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1
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Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
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– 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.
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– DanielWainfleet
Jan 3 at 10:26
add a comment |
$begingroup$
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
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
$endgroup$
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
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
asked Dec 27 '18 at 16:36
Keefer RowanKeefer Rowan
21010
21010
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
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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
add a comment |
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
add a comment |
1 Answer
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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.
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
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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.
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
$endgroup$
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.
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
answered Jan 3 at 4:34
Alex RavskyAlex Ravsky
43.3k32583
43.3k32583
add a comment |
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Yes, the question makes sense. The answer to it is "no". See this: math.stackexchange.com/questions/20913/…
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– 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