homeomorphic to [-1,1]
It's written that one point compactification of $[−1,1]setminus{0}$ is homeomorphic to $[-1,1]$ in my topology book. I want to know why they are homeomorphic.
general-topology
asked Nov 25 at 7:45
dlfjsemf
968
add a comment |
It's written that one point compactification of $[−1,1]setminus{0}$ is homeomorphic to $[-1,1]$ in my topology book. I want to know why they are homeomorphic.
general-topology
asked Nov 25 at 7:45
dlfjsemf
968
add a comment |
It's written that one point compactification of $[−1,1]setminus{0}$ is homeomorphic to $[-1,1]$ in my topology book. I want to know why they are homeomorphic.
general-topology
asked Nov 25 at 7:45
dlfjsemf
968
It's written that one point compactification of $[−1,1]setminus{0}$ is homeomorphic to $[-1,1]$ in my topology book. I want to know why they are homeomorphic.
general-topology
general-topology
asked Nov 25 at 7:45
dlfjsemf
968
asked Nov 25 at 7:45
dlfjsemf
968
edited Nov 25 at 9:39
asked Nov 25 at 7:45
dlfjsemf
968
asked Nov 25 at 7:45
dlfjsemf
968
asked Nov 25 at 7:45
dlfjsemf
968
968
add a comment |
add a comment |
2 Answers
2
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oldest
votes
Fact:
If $X$ is locally compact Hausdorff and we have a compact Hausdorff space $Y$ such
that $X subseteq Y$ and $Ysetminus X$ consists of a single point $p$, then $Y$ is homeomorphic to the one-point compactification $alpha X$ of $X$.
Apply this to $X=[-1,1]setminus{0}$ and $Y=[-1,1]$.
To prove the fact, let $alpha X = X cup {infty}$ in its usual topology.
Check that $f: Y to alpha X$ defined by $f(x)=x$ for $x in X$ and $f(infty) = p$ is a continuous bijection and then compactness (and Hausdorffness) makes this a homeomorphism.
A slight generalisation of the fact: if $Y$ is compact Hausdorff and for some point $p in Y$ we have that $Ysetminus{p}$ is homeomorphic to $X$, then $Y$ is homeomorphic to the one-point compactifcation of $X$. I.e. if $Y$ is compact and one point larger than something homeomorphic to $X$, it must be its one-point compactification. It shows this compactification is essentially unique.
answered Nov 25 at 8:41
Henno Brandsma
104k346113
add a comment |
No, they are not homeomorphic, the second set is connected, but not the first.
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
add a comment |
Your Answer
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2 Answers
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2 Answers
2
active
oldest
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active
oldest
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active
oldest
votes
Fact:
If $X$ is locally compact Hausdorff and we have a compact Hausdorff space $Y$ such
that $X subseteq Y$ and $Ysetminus X$ consists of a single point $p$, then $Y$ is homeomorphic to the one-point compactification $alpha X$ of $X$.
Apply this to $X=[-1,1]setminus{0}$ and $Y=[-1,1]$.
To prove the fact, let $alpha X = X cup {infty}$ in its usual topology.
Check that $f: Y to alpha X$ defined by $f(x)=x$ for $x in X$ and $f(infty) = p$ is a continuous bijection and then compactness (and Hausdorffness) makes this a homeomorphism.
A slight generalisation of the fact: if $Y$ is compact Hausdorff and for some point $p in Y$ we have that $Ysetminus{p}$ is homeomorphic to $X$, then $Y$ is homeomorphic to the one-point compactifcation of $X$. I.e. if $Y$ is compact and one point larger than something homeomorphic to $X$, it must be its one-point compactification. It shows this compactification is essentially unique.
answered Nov 25 at 8:41
Henno Brandsma
104k346113
add a comment |
Fact:
If $X$ is locally compact Hausdorff and we have a compact Hausdorff space $Y$ such
that $X subseteq Y$ and $Ysetminus X$ consists of a single point $p$, then $Y$ is homeomorphic to the one-point compactification $alpha X$ of $X$.
Apply this to $X=[-1,1]setminus{0}$ and $Y=[-1,1]$.
To prove the fact, let $alpha X = X cup {infty}$ in its usual topology.
Check that $f: Y to alpha X$ defined by $f(x)=x$ for $x in X$ and $f(infty) = p$ is a continuous bijection and then compactness (and Hausdorffness) makes this a homeomorphism.
A slight generalisation of the fact: if $Y$ is compact Hausdorff and for some point $p in Y$ we have that $Ysetminus{p}$ is homeomorphic to $X$, then $Y$ is homeomorphic to the one-point compactifcation of $X$. I.e. if $Y$ is compact and one point larger than something homeomorphic to $X$, it must be its one-point compactification. It shows this compactification is essentially unique.
answered Nov 25 at 8:41
Henno Brandsma
104k346113
add a comment |
Fact:
If $X$ is locally compact Hausdorff and we have a compact Hausdorff space $Y$ such
that $X subseteq Y$ and $Ysetminus X$ consists of a single point $p$, then $Y$ is homeomorphic to the one-point compactification $alpha X$ of $X$.
Apply this to $X=[-1,1]setminus{0}$ and $Y=[-1,1]$.
To prove the fact, let $alpha X = X cup {infty}$ in its usual topology.
Check that $f: Y to alpha X$ defined by $f(x)=x$ for $x in X$ and $f(infty) = p$ is a continuous bijection and then compactness (and Hausdorffness) makes this a homeomorphism.
A slight generalisation of the fact: if $Y$ is compact Hausdorff and for some point $p in Y$ we have that $Ysetminus{p}$ is homeomorphic to $X$, then $Y$ is homeomorphic to the one-point compactifcation of $X$. I.e. if $Y$ is compact and one point larger than something homeomorphic to $X$, it must be its one-point compactification. It shows this compactification is essentially unique.
answered Nov 25 at 8:41
Henno Brandsma
104k346113
Fact:
If $X$ is locally compact Hausdorff and we have a compact Hausdorff space $Y$ such
that $X subseteq Y$ and $Ysetminus X$ consists of a single point $p$, then $Y$ is homeomorphic to the one-point compactification $alpha X$ of $X$.
Apply this to $X=[-1,1]setminus{0}$ and $Y=[-1,1]$.
To prove the fact, let $alpha X = X cup {infty}$ in its usual topology.
Check that $f: Y to alpha X$ defined by $f(x)=x$ for $x in X$ and $f(infty) = p$ is a continuous bijection and then compactness (and Hausdorffness) makes this a homeomorphism.
A slight generalisation of the fact: if $Y$ is compact Hausdorff and for some point $p in Y$ we have that $Ysetminus{p}$ is homeomorphic to $X$, then $Y$ is homeomorphic to the one-point compactifcation of $X$. I.e. if $Y$ is compact and one point larger than something homeomorphic to $X$, it must be its one-point compactification. It shows this compactification is essentially unique.
answered Nov 25 at 8:41
Henno Brandsma
104k346113
answered Nov 25 at 8:41
Henno Brandsma
104k346113
answered Nov 25 at 8:41
Henno Brandsma
104k346113
answered Nov 25 at 8:41
Henno Brandsma
104k346113
104k346113
add a comment |
add a comment |
No, they are not homeomorphic, the second set is connected, but not the first.
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
add a comment |
No, they are not homeomorphic, the second set is connected, but not the first.
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
add a comment |
No, they are not homeomorphic, the second set is connected, but not the first.
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
No, they are not homeomorphic, the second set is connected, but not the first.
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
answered Nov 25 at 7:48
Tsemo Aristide
55.4k11444
55.4k11444
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
add a comment |
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
It's written that one compactification of $[-1,1]setminus{0}$ is homeomorphic to [-1,1] in my topology book. Is it wrong?
– dlfjsemf
Nov 25 at 8:01
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
@dlfjsemf no it's correct.
– Henno Brandsma
Nov 25 at 8:35
add a comment |
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