Path connectedness

A path-connected space is a topological space $(𝑋, T)$ in which all points $𝑥, 𝑦 \in 𝑋$ may be connected by a Continuous path $𝑐 : [0, 1] \to 𝑋$ so that $𝑐 (0) = 𝑥$ and $𝑐 (1) = 𝑦$ . A subset of $𝑋$ may also be path-connected. #m/def/topology

If $𝑐$ is a continuous arc then $𝑋$ is called arc-connected. Path connectedness is stronger than ordinary connectedness (Path connected implies connected), but weaker than arc connectedness. Not every path-connected space is arc-connected, take the Line with two origins.

Path-connected components

Two points $𝑥, 𝑦 \in 𝑋$ are said to be path-connected iff there exists a connected path between them, and we write $𝑥 \sim 𝑦$ . This is an equivalence relation (Connectedness is transitive) and the equivalence classes are called path-connected components of $𝑋$ .

Properties

Main theorem of connectedness: Path connectedness is preserved by continuous functions, thus it is a Topological property.
Connectedness is transitive: Path-connected neighbourhoods of a point have a path-connected union.
Any $𝑐 ([0, 1])$ is connected
Convex subsets of $ℝ 𝑛$ are connected
Every path-connected Hausdorff space is arc-connected.
A space is path-connected iff constant maps form a homotopy class.
Path connectedness is homotopy invariant

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