Connectedness

Path connectedness

A path-connected space is a topological space (𝑋,T) in which all points 𝑥,𝑦 ∈𝑋 may be connected by a Continuous path 𝑐 :[0,1] →𝑋 so that 𝑐(0) =𝑥 and 𝑐(1) =𝑦. A subset of 𝑋 may also be path-connected. 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 ∈𝑋 are said to be path-connected iff there exists a connected path between them, and we write 𝑥 ∼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 ℝ𝑛 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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