Homeomorphic characterization of the real line?
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
27.1k380132
asked Jan 1 at 12:07
James Baxter
23612
add a comment |
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
27.1k380132
asked Jan 1 at 12:07
James Baxter
23612
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
Possible duplicate of Topological Characterisation of the real line.
– Chris Godsil
8 hours ago
add a comment |
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
27.1k380132
asked Jan 1 at 12:07
James Baxter
23612
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
gn.general-topology gt.geometric-topology
edited Jan 1 at 12:34
YCor
27.1k380132
asked Jan 1 at 12:07
James Baxter
23612
edited Jan 1 at 12:34
YCor
27.1k380132
asked Jan 1 at 12:07
James Baxter
23612
edited Jan 1 at 12:34
YCor
27.1k380132
edited Jan 1 at 12:34
YCor
27.1k380132
edited Jan 1 at 12:34
YCor
27.1k380132
27.1k380132
asked Jan 1 at 12:07
James Baxter
23612
asked Jan 1 at 12:07
James Baxter
23612
asked Jan 1 at 12:07
James Baxter
23612
23612
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
Possible duplicate of Topological Characterisation of the real line.
– Chris Godsil
8 hours ago
add a comment |
3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
Possible duplicate of Topological Characterisation of the real line.
– Chris Godsil
8 hours ago
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
Possible duplicate of Topological Characterisation of the real line.
– Chris Godsil
8 hours ago
Possible duplicate of Topological Characterisation of the real line.
– Chris Godsil
8 hours ago
add a comment |
1 Answer
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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
user131781
27613
add a comment |
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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
user131781
27613
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
user131781
27613
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
user131781
27613
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
user131781
27613
answered Jan 1 at 12:30
user131781
27613
answered Jan 1 at 12:30
user131781
27613
answered Jan 1 at 12:30
user131781
27613
27613
add a comment |
add a comment |
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3
related post: mathoverflow.net/questions/76134/…
– Josiah Park
Jan 1 at 12:33
Possible duplicate of Topological Characterisation of the real line.
– Chris Godsil
8 hours ago