Path-connected topological space

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

Definition

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

• $\forall x_1,x_2\in X\exists p\in$$C([0,1],X)$$[p(0)\eq x_1\wedge p(1)\eq x_2]$
  • In words: for all points in $X$ 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