# Path-connected topological space

From Maths

(Redirected from Path connected (topology))

**Stub grade: A***

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Important to cover from multiple sources, demote after this.

Find home for:

- 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)

## Contents

## 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^{[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.

## 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