[[Local (path) connectedness]]
# Locally path connected spaces have identical connected and path-connected components

Let $X$ be a [[Local (path) connectedness|locally path-connected]] topological space.
Then for any $a,b \in X$,
$a$ is [[Path connectedness|path-connected]] to $b$
iff $a$ is [[Connectedness|connected]] to $b$. #m/thm/topology 

> [!missing]- 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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#state/develop | #lang/en | #SemBr