Local (path) connectedness

Locally path connected spaces have identical connected and path-connected components

Let 𝑋 be a locally path-connected topological space. Then for any 𝑎,𝑏 ∈𝑋, 𝑎 is path-connected to 𝑏 iff 𝑎 is connected to 𝑏. topology

Proof

proof See for example @looseAlgebraischeTopologie2010, p. 46

Corollaries

• A locally path-connected space is path-connected iff it is connected

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