Homeomorphic characterization of the real line? [duplicate]












10












$begingroup$



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







share|cite|improve this question











$endgroup$



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




    $begingroup$
    related post: mathoverflow.net/questions/76134/…
    $endgroup$
    – Josiah Park
    Jan 1 at 12:33
















10












$begingroup$



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







share|cite|improve this question











$endgroup$



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




    $begingroup$
    related post: mathoverflow.net/questions/76134/…
    $endgroup$
    – Josiah Park
    Jan 1 at 12:33














10












10








10


1



$begingroup$



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







share|cite|improve this question











$endgroup$





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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 1 at 12:34









YCor

27.7k482134




27.7k482134










asked Jan 1 at 12:07









James BaxterJames Baxter

29613




29613




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




    $begingroup$
    related post: mathoverflow.net/questions/76134/…
    $endgroup$
    – Josiah Park
    Jan 1 at 12:33














  • 3




    $begingroup$
    related post: mathoverflow.net/questions/76134/…
    $endgroup$
    – Josiah Park
    Jan 1 at 12:33








3




3




$begingroup$
related post: mathoverflow.net/questions/76134/…
$endgroup$
– Josiah Park
Jan 1 at 12:33




$begingroup$
related post: mathoverflow.net/questions/76134/…
$endgroup$
– Josiah Park
Jan 1 at 12:33










1 Answer
1






active

oldest

votes


















20












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
    $endgroup$
    – Taras Banakh
    Jan 5 at 21:35


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









20












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
    $endgroup$
    – Taras Banakh
    Jan 5 at 21:35
















20












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
    $endgroup$
    – Taras Banakh
    Jan 5 at 21:35














20












20








20





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 1 at 12:30









user131781user131781

32613




32613












  • $begingroup$
    In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
    $endgroup$
    – Taras Banakh
    Jan 5 at 21:35


















  • $begingroup$
    In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
    $endgroup$
    – Taras Banakh
    Jan 5 at 21:35
















$begingroup$
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
$endgroup$
– Taras Banakh
Jan 5 at 21:35




$begingroup$
In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact?
$endgroup$
– Taras Banakh
Jan 5 at 21:35



-

Popular posts from this blog

Monofisismo

Image
Monofisismo (do grego: monos - "único, singular" e physis - "natureza") é o ponto de vista cristológico que defende que, depois da união do divino e do humano na encarnação histórica, Jesus Cristo, como encarnação do Filho ou Verbo ( Logos ) de Deus, teria apenas uma única "natureza", a divina, e não uma síntese de ambas. O monofisismo é contraposto pelo diofisismo (ou "diafisismo"), que defende que Jesus preservou em si as duas naturezas. Historicamente, o monofisismo se refere primordialmente à posição dos que (especialmente no Egito e, em menor grau, na Síria) rejeitaram as decisões do Concílio de Calcedônia em 451 (o quarto concílio ecumênico). Os membros mais moderados entre eles, porém, defendem a teologia "miafisita", que se tornou a oficial para as Igrejas Ortodoxas Orientais. Muitos ortodoxos orientais, porém, rejeitam essa classificação, mesmo como um termo genérico, mas ele é amplamente utilizado na literatura históri
Read more

Angular Downloading a file using contenturl with Basic Authentication

Image
0 I have a contentUrl when user clicks on that,file should be downloaded. If I try to browse the contentUrl in the browser it asks for Userid and password.I believe this is Basic Authentication.I dont want user to enter UserId and Password so I am passing it Authorization(Please see below code). Url looks some thing like this: http://XXX.XXX.XX.XXX:8080/flowable-rest/service/runtime/tasks/90875/attachments/90555/content let username : string = 'admin'; let pwd : string = 'test'; let headers = new Headers(); headers.set('Accept', 'text/json'); headers.append('Content-Type', 'application/json'); headers.set('Access-Control-Allow-Origin', '*'); headers.set('Access-Control-Allow-Methods','POST, GET, OPTIONS, PUT, DELETE'); head
Read more

Olmecas

Image
Monumento 1, uma das quatro cabeças colossais encontradas em La Venta, com altura aproximada de 3 metros Olmecas é a designação do povo e da civilização que estiveram na origem da antiga cultura pré-colombiana da Mesoamérica e que se desenvolveram nas regiões tropicais do centro-sul do atual México durante o pré-clássico, próximo de onde hoje estão localizados os estados mexicanos de Veracruz e Tabasco, no Istmo de Tehuantepec, numa zona designada área nuclear olmeca. A cultura olmeca floresceu nesta região aproximadamente entre 1500 e 400 a.C., [ 1 ] e crê-se que tenha sido a civilização-mãe de todas as civilizações mesoamericanas que se desenvolveram posteriormente. [ 2 ] No entanto, desconhece-se a sua exacta filiação étnica, ainda que existam numerosas hipóteses colocadas para tentar resolver esta questão. O etnónimo olmeca foi cunhado pelos arqueólogos do século XX, e não devem confundir-se com os muito posteriores olmecas-xicalancas que ocuparam vários locais do México
Read more