Open connected subsets of path connected spaces

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Does every open and connected subset of path connected topological space has to be path connected? Statement should be false as there is a similar theorem but for Euclidean spaces, however I can't think of a counterexample. What about the same statement, but for path connected metric spaces?






general-topology examples-counterexamples connectedness path-connected







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Does every open and connected subset of path connected topological space has to be path connected? Statement should be false as there is a similar theorem but for Euclidean spaces, however I can't think of a counterexample. What about the same statement, but for path connected metric spaces?






general-topology examples-counterexamples connectedness path-connected







share|cite|improve this question





















  • math.stackexchange.com/questions/766422/…
    – John Douma
    5 hours ago












up vote
4
down vote

favorite









up vote
4
down vote

favorite











Does every open and connected subset of path connected topological space has to be path connected? Statement should be false as there is a similar theorem but for Euclidean spaces, however I can't think of a counterexample. What about the same statement, but for path connected metric spaces?






general-topology examples-counterexamples connectedness path-connected







share|cite|improve this question













Does every open and connected subset of path connected topological space has to be path connected? Statement should be false as there is a similar theorem but for Euclidean spaces, however I can't think of a counterexample. What about the same statement, but for path connected metric spaces?






general-topology examples-counterexamples connectedness path-connected




general-topology examples-counterexamples connectedness path-connected






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Uros Dinic

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  • math.stackexchange.com/questions/766422/…
    – John Douma
    5 hours ago
















  • math.stackexchange.com/questions/766422/…
    – John Douma
    5 hours ago















math.stackexchange.com/questions/766422/…
– John Douma
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math.stackexchange.com/questions/766422/…
– John Douma
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The classical example of a connected (metric) space that is not path-connected is the topologist's sine curve. I will give an example based on this.



Consider the graph of the $sin(frac1x)$ function on $(0,1]$.
enter image description here



Let $X$ be the space which consists of this graph together with the vertical line segment connecting $(0,-1)$ and $(0,1)$, and the curve in red:enter image description here



$X$ is a metric space that is path connected. You can also clearly see this as an open subset of $X$:



enter image description here



This is an open connected subspace of the $X$ that is not path connected.



This counter example is a metric space. It applies as well for the general case of topological spaces.






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    Consider the space $$X=left(x,y)inBbb R^2,:, (x=0land yle 2)lor left(xne 0land y=sinfrac1xright) lor (yge 0land x^2+y^2=4)right$$



    I.e. a topologist sine, plus an appropriate vertical half-line, plus a half circle "path-connecting" the curve to the tip of the half-line. Then, $Xsetminus (0,2)$ is connected, but not path-connected.






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      The classical example of a connected (metric) space that is not path-connected is the topologist's sine curve. I will give an example based on this.



      Consider the graph of the $sin(frac1x)$ function on $(0,1]$.
      enter image description here



      Let $X$ be the space which consists of this graph together with the vertical line segment connecting $(0,-1)$ and $(0,1)$, and the curve in red:enter image description here



      $X$ is a metric space that is path connected. You can also clearly see this as an open subset of $X$:



      enter image description here



      This is an open connected subspace of the $X$ that is not path connected.



      This counter example is a metric space. It applies as well for the general case of topological spaces.






      share|cite|improve this answer
























        up vote
        3
        down vote













        The classical example of a connected (metric) space that is not path-connected is the topologist's sine curve. I will give an example based on this.



        Consider the graph of the $sin(frac1x)$ function on $(0,1]$.
        enter image description here



        Let $X$ be the space which consists of this graph together with the vertical line segment connecting $(0,-1)$ and $(0,1)$, and the curve in red:enter image description here



        $X$ is a metric space that is path connected. You can also clearly see this as an open subset of $X$:



        enter image description here



        This is an open connected subspace of the $X$ that is not path connected.



        This counter example is a metric space. It applies as well for the general case of topological spaces.






        share|cite|improve this answer






















          up vote
          3
          down vote










          up vote
          3
          down vote









          The classical example of a connected (metric) space that is not path-connected is the topologist's sine curve. I will give an example based on this.



          Consider the graph of the $sin(frac1x)$ function on $(0,1]$.
          enter image description here



          Let $X$ be the space which consists of this graph together with the vertical line segment connecting $(0,-1)$ and $(0,1)$, and the curve in red:enter image description here



          $X$ is a metric space that is path connected. You can also clearly see this as an open subset of $X$:



          enter image description here



          This is an open connected subspace of the $X$ that is not path connected.



          This counter example is a metric space. It applies as well for the general case of topological spaces.






          share|cite|improve this answer












          The classical example of a connected (metric) space that is not path-connected is the topologist's sine curve. I will give an example based on this.



          Consider the graph of the $sin(frac1x)$ function on $(0,1]$.
          enter image description here



          Let $X$ be the space which consists of this graph together with the vertical line segment connecting $(0,-1)$ and $(0,1)$, and the curve in red:enter image description here



          $X$ is a metric space that is path connected. You can also clearly see this as an open subset of $X$:



          enter image description here



          This is an open connected subspace of the $X$ that is not path connected.



          This counter example is a metric space. It applies as well for the general case of topological spaces.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 5 hours ago









          Scientifica

          5,66631331




          5,66631331




















              up vote
              2
              down vote













              Consider the space $$X=left(x,y)inBbb R^2,:, (x=0land yle 2)lor left(xne 0land y=sinfrac1xright) lor (yge 0land x^2+y^2=4)right$$



              I.e. a topologist sine, plus an appropriate vertical half-line, plus a half circle "path-connecting" the curve to the tip of the half-line. Then, $Xsetminus (0,2)$ is connected, but not path-connected.






              share|cite|improve this answer
























                up vote
                2
                down vote













                Consider the space $$X=left(x,y)inBbb R^2,:, (x=0land yle 2)lor left(xne 0land y=sinfrac1xright) lor (yge 0land x^2+y^2=4)right$$



                I.e. a topologist sine, plus an appropriate vertical half-line, plus a half circle "path-connecting" the curve to the tip of the half-line. Then, $Xsetminus (0,2)$ is connected, but not path-connected.






                share|cite|improve this answer






















                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  Consider the space $$X=left(x,y)inBbb R^2,:, (x=0land yle 2)lor left(xne 0land y=sinfrac1xright) lor (yge 0land x^2+y^2=4)right$$



                  I.e. a topologist sine, plus an appropriate vertical half-line, plus a half circle "path-connecting" the curve to the tip of the half-line. Then, $Xsetminus (0,2)$ is connected, but not path-connected.






                  share|cite|improve this answer












                  Consider the space $$X=left(x,y)inBbb R^2,:, (x=0land yle 2)lor left(xne 0land y=sinfrac1xright) lor (yge 0land x^2+y^2=4)right$$



                  I.e. a topologist sine, plus an appropriate vertical half-line, plus a half circle "path-connecting" the curve to the tip of the half-line. Then, $Xsetminus (0,2)$ is connected, but not path-connected.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 5 hours ago









                  Saucy O'Path

                  4,9791424




                  4,9791424



























                       

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