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)

Definition

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:

• [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.

See next

• If a topological space is path-connected then it is connected - the point of this really
• The image of a path-connected space under a continuous map is also path-connected
• Given an arbitrary collection of path-connected components that one point in common their union is path-connected
• The product of finitely many path-connected spaces is path connected
• Every quotient space of a path-connected space is path-connected

See also

• Locally path-connected