Mixing
Homeomorphic characterization of the real line? [duplicate]
This question already has an answer here:
Topological Characterisation of the real line.
4 answers
Let $A$ be a path-connected subset of $mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.
Is $A$ necessarily homeomorphic to $mathbb{R}$?
gn.general-topology gt.geometric-topology
edited Jan 1 at 12:34
YCor
27.2k480132
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
marked as duplicate by Lee Mosher, Chris Godsil, Ben McKay, Community♦ Jan 8 at 16:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
This question already has an answer here:
Topological Characterisation of the real line.
4 answers
Let $A$ be a path-connected subset of $mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.
Is $A$ necessarily homeomorphic to $mathbb{R}$?
gn.general-topology gt.geometric-topology
edited Jan 1 at 12:34
YCor
27.2k480132
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
marked as duplicate by Lee Mosher, Chris Godsil, Ben McKay, Community♦ Jan 8 at 16:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
add a comment |
This question already has an answer here:
Topological Characterisation of the real line.
4 answers
Let $A$ be a path-connected subset of $mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.
Is $A$ necessarily homeomorphic to $mathbb{R}$?
gn.general-topology gt.geometric-topology
edited Jan 1 at 12:34
YCor
27.2k480132
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
This question already has an answer here:
Topological Characterisation of the real line.
4 answers
Let $A$ be a path-connected subset of $mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.
Is $A$ necessarily homeomorphic to $mathbb{R}$?
This question already has an answer here:
Topological Characterisation of the real line.
4 answers
gn.general-topology gt.geometric-topology
gn.general-topology gt.geometric-topology
edited Jan 1 at 12:34
YCor
27.2k480132
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
edited Jan 1 at 12:34
YCor
27.2k480132
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
edited Jan 1 at 12:34
YCor
27.2k480132
edited Jan 1 at 12:34
YCor
27.2k480132
edited Jan 1 at 12:34
YCor
27.2k480132
27.2k480132
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
asked Jan 1 at 12:07
James BaxterJames Baxter
27812
27812
marked as duplicate by Lee Mosher, Chris Godsil, Ben McKay, Community♦ Jan 8 at 16:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Lee Mosher, Chris Godsil, Ben McKay, Community♦ Jan 8 at 16:59
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
add a comment |
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
3
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
add a comment |
1 Answer
1
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Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has more than one point.
answered Jan 1 at 12:30
user131781user131781
28613
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has more than one point.
answered Jan 1 at 12:30
user131781user131781
28613
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
add a comment |
Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has more than one point.
answered Jan 1 at 12:30
user131781user131781
28613
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
add a comment |
Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has more than one point.
answered Jan 1 at 12:30
user131781user131781
28613
Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has more than one point.
answered Jan 1 at 12:30
user131781user131781
28613
answered Jan 1 at 12:30
user131781user131781
28613
answered Jan 1 at 12:30
user131781user131781
28613
answered Jan 1 at 12:30
user131781user131781
28613
28613
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
add a comment |
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
– Taras Banakh
Jan 5 at 21:35
add a comment |
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33