Path-connected topological space

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  • The n-torus, [ilmath]\mathbb{T}^n[/ilmath] is path connected as it is a finite product of circles
  • Any convex subset of [ilmath]\mathbb{R}^n[/ilmath] is path connected.
  • [ilmath]\mathbb{R}^n-\{0\} [/ilmath] is path-connected for [ilmath]n\ge 2[/ilmath]
  • The [ilmath]n[/ilmath]-sphere for [ilmath]n\ge 1[/ilmath]- by quotient space definition really (which is what again) Alec (talk) 12:52, 23 February 2017 (UTC)


Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, we say that [ilmath]X[/ilmath] is path connected or is a path connected (topological) space if the space has the following property[1]:

  • [ilmath]\forall x_1,x_2\in X\exists p\in [/ilmath][ilmath]C([0,1],X)[/ilmath][ilmath][p(0)\eq x_1\wedge p(1)\eq x_2][/ilmath]
    • In words: for all points in [ilmath]X[/ilmath] there exists a path (notice that it's a path in the topological sense) that starts at one of the points and ends at another.

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See also


  1. Introduction to Topological Manifolds - John M. Lee