Homeomorphism between [0,1]/~ and the Hawaiian Earring
Let $X$ be the quotient space [0,1]/~ where 0 ~ 1 ~ 1/2 ~ $cdots$ ~ 1/n ~ $cdots$
Let $H$ (the Hawaiian Earring) be the subspace of $mathbb{R}^2$ consisting of the union of circles of radius 1/n and centered at (0,1/n) for all n=1,2,3,...
I need to find a homeomorphism between X and H.
Note: I am studying general topology, so I can't use algebraic topology.
general-topology quotient-spaces
asked Dec 9 at 19:11
111
275
add a comment |
Let $X$ be the quotient space [0,1]/~ where 0 ~ 1 ~ 1/2 ~ $cdots$ ~ 1/n ~ $cdots$
Let $H$ (the Hawaiian Earring) be the subspace of $mathbb{R}^2$ consisting of the union of circles of radius 1/n and centered at (0,1/n) for all n=1,2,3,...
I need to find a homeomorphism between X and H.
Note: I am studying general topology, so I can't use algebraic topology.
general-topology quotient-spaces
asked Dec 9 at 19:11
111
275
What have you tried so far?
– Guus Palmer
Dec 9 at 19:15
I'm having trouble figuring out how to send a point x from [0,1] to a circle of radius x.
– 111
Dec 9 at 19:17
Your question is not related to algebraic topology!
– Paul Frost
Dec 9 at 23:08
Sorry I'm new to topology and I just wanted to make sure that any answers did not employ algebraic techniques, sorry
– 111
Dec 9 at 23:10
1
There is no reason to be sorry, your question is fine. I just wanted to say that it is a question to be treated in general topology.
– Paul Frost
Dec 9 at 23:13
add a comment |
Let $X$ be the quotient space [0,1]/~ where 0 ~ 1 ~ 1/2 ~ $cdots$ ~ 1/n ~ $cdots$
Let $H$ (the Hawaiian Earring) be the subspace of $mathbb{R}^2$ consisting of the union of circles of radius 1/n and centered at (0,1/n) for all n=1,2,3,...
I need to find a homeomorphism between X and H.
Note: I am studying general topology, so I can't use algebraic topology.
general-topology quotient-spaces
asked Dec 9 at 19:11
111
275
Let $X$ be the quotient space [0,1]/~ where 0 ~ 1 ~ 1/2 ~ $cdots$ ~ 1/n ~ $cdots$
Let $H$ (the Hawaiian Earring) be the subspace of $mathbb{R}^2$ consisting of the union of circles of radius 1/n and centered at (0,1/n) for all n=1,2,3,...
I need to find a homeomorphism between X and H.
Note: I am studying general topology, so I can't use algebraic topology.
general-topology quotient-spaces
general-topology quotient-spaces
asked Dec 9 at 19:11
111
275
asked Dec 9 at 19:11
111
275
asked Dec 9 at 19:11
111
275
asked Dec 9 at 19:11
111
275
asked Dec 9 at 19:11
111
275
275
What have you tried so far?
– Guus Palmer
Dec 9 at 19:15
I'm having trouble figuring out how to send a point x from [0,1] to a circle of radius x.
– 111
Dec 9 at 19:17
Your question is not related to algebraic topology!
– Paul Frost
Dec 9 at 23:08
Sorry I'm new to topology and I just wanted to make sure that any answers did not employ algebraic techniques, sorry
– 111
Dec 9 at 23:10
1
There is no reason to be sorry, your question is fine. I just wanted to say that it is a question to be treated in general topology.
– Paul Frost
Dec 9 at 23:13
add a comment |
What have you tried so far?
– Guus Palmer
Dec 9 at 19:15
I'm having trouble figuring out how to send a point x from [0,1] to a circle of radius x.
– 111
Dec 9 at 19:17
Your question is not related to algebraic topology!
– Paul Frost
Dec 9 at 23:08
Sorry I'm new to topology and I just wanted to make sure that any answers did not employ algebraic techniques, sorry
– 111
Dec 9 at 23:10
1
There is no reason to be sorry, your question is fine. I just wanted to say that it is a question to be treated in general topology.
– Paul Frost
Dec 9 at 23:13
What have you tried so far?
– Guus Palmer
Dec 9 at 19:15
What have you tried so far?
– Guus Palmer
Dec 9 at 19:15
I'm having trouble figuring out how to send a point x from [0,1] to a circle of radius x.
– 111
Dec 9 at 19:17
I'm having trouble figuring out how to send a point x from [0,1] to a circle of radius x.
– 111
Dec 9 at 19:17
Your question is not related to algebraic topology!
– Paul Frost
Dec 9 at 23:08
Your question is not related to algebraic topology!
– Paul Frost
Dec 9 at 23:08
Sorry I'm new to topology and I just wanted to make sure that any answers did not employ algebraic techniques, sorry
– 111
Dec 9 at 23:10
Sorry I'm new to topology and I just wanted to make sure that any answers did not employ algebraic techniques, sorry
– 111
Dec 9 at 23:10
1
1
There is no reason to be sorry, your question is fine. I just wanted to say that it is a question to be treated in general topology.
– Paul Frost
Dec 9 at 23:13
There is no reason to be sorry, your question is fine. I just wanted to say that it is a question to be treated in general topology.
– Paul Frost
Dec 9 at 23:13
add a comment |
1 Answer
1
active
oldest
votes
You just map $left[frac12,1right]$ onto the largest circle, $left[frac13,frac12right]$ onto the largest one and so on…
That is, if $xin(0,1]$, then $x=fraclambda{n+1}+frac{1-lambda}n$ for some $ninmathbb N$ and some $lambdain[0,1]$. Then$$f(x)=left(frac1nsin(2pilambda),frac1n+-frac1ncos(2pilambda)right).$$And, of course, $f(0)=(0,0)$.
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
What is the motivation for this formula?
– 111
Dec 9 at 22:09
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
1
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
1
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
1
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
|
show 8 more comments
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1 Answer
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You just map $left[frac12,1right]$ onto the largest circle, $left[frac13,frac12right]$ onto the largest one and so on…
That is, if $xin(0,1]$, then $x=fraclambda{n+1}+frac{1-lambda}n$ for some $ninmathbb N$ and some $lambdain[0,1]$. Then$$f(x)=left(frac1nsin(2pilambda),frac1n+-frac1ncos(2pilambda)right).$$And, of course, $f(0)=(0,0)$.
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
What is the motivation for this formula?
– 111
Dec 9 at 22:09
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
1
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
1
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
1
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
|
show 8 more comments
You just map $left[frac12,1right]$ onto the largest circle, $left[frac13,frac12right]$ onto the largest one and so on…
That is, if $xin(0,1]$, then $x=fraclambda{n+1}+frac{1-lambda}n$ for some $ninmathbb N$ and some $lambdain[0,1]$. Then$$f(x)=left(frac1nsin(2pilambda),frac1n+-frac1ncos(2pilambda)right).$$And, of course, $f(0)=(0,0)$.
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
What is the motivation for this formula?
– 111
Dec 9 at 22:09
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
1
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
1
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
1
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
|
show 8 more comments
You just map $left[frac12,1right]$ onto the largest circle, $left[frac13,frac12right]$ onto the largest one and so on…
That is, if $xin(0,1]$, then $x=fraclambda{n+1}+frac{1-lambda}n$ for some $ninmathbb N$ and some $lambdain[0,1]$. Then$$f(x)=left(frac1nsin(2pilambda),frac1n+-frac1ncos(2pilambda)right).$$And, of course, $f(0)=(0,0)$.
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
You just map $left[frac12,1right]$ onto the largest circle, $left[frac13,frac12right]$ onto the largest one and so on…
That is, if $xin(0,1]$, then $x=fraclambda{n+1}+frac{1-lambda}n$ for some $ninmathbb N$ and some $lambdain[0,1]$. Then$$f(x)=left(frac1nsin(2pilambda),frac1n+-frac1ncos(2pilambda)right).$$And, of course, $f(0)=(0,0)$.
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
edited Dec 9 at 22:45
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
answered Dec 9 at 19:19
José Carlos Santos
150k22121221
150k22121221
What is the motivation for this formula?
– 111
Dec 9 at 22:09
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
1
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
1
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
1
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
|
show 8 more comments
What is the motivation for this formula?
– 111
Dec 9 at 22:09
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
1
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
1
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
1
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
What is the motivation for this formula?
– 111
Dec 9 at 22:09
What is the motivation for this formula?
– 111
Dec 9 at 22:09
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
That's a strange question. That's what occured to me to solve the problem.
– José Carlos Santos
Dec 9 at 22:16
1
1
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
You are right: you stated it from the start. I shall edit my answer then.
– José Carlos Santos
Dec 9 at 22:40
1
1
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
I've edited my answer. I hope that everything is clear now.
– José Carlos Santos
Dec 9 at 22:45
1
1
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
For continuity, see $f$ as a map from $[0,1]$ onto $H$. It is clear that $f$ is continuous on each interval $left[frac1{n+1},frac1nright]$. So, it is continuous on $(0,1]$. FInally, it is clear that $f(0)=lim_{xto0}f(x)$ and that therefore $f$ is continuous on $0$ too. Then, yes, just use the fact that $[0,1]$ is compact.
– José Carlos Santos
Dec 9 at 23:25
|
show 8 more comments
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What have you tried so far?
– Guus Palmer
Dec 9 at 19:15
I'm having trouble figuring out how to send a point x from [0,1] to a circle of radius x.
– 111
Dec 9 at 19:17
Your question is not related to algebraic topology!
– Paul Frost
Dec 9 at 23:08
Sorry I'm new to topology and I just wanted to make sure that any answers did not employ algebraic techniques, sorry
– 111
Dec 9 at 23:10
1
There is no reason to be sorry, your question is fine. I just wanted to say that it is a question to be treated in general topology.
– Paul Frost
Dec 9 at 23:13