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

Let $𝑋$ be a locally path-connected topological space. Then for any $𝑎, 𝑏 \in 𝑋$ , $𝑎$ is path-connected to $𝑏$ iff $𝑎$ is connected to $𝑏$ . #m/thm/topology

Proof

#missing/proof See for example @looseAlgebraischeTopologie2010, p. 46

Corollaries

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

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