[[Connectedness]] and [[Path connectedness]]
# Local (path) connectedness

A topological space $X$ is **(path-)connected** iff for each $x \in X$ and every neighbourhood $U$ of $X$,
there exists a (path-)connected **open** neighbourhood $V$ of $x$
such that $V \sube U$. 
Equivalently, every point has a [[neighbourhood basis]] of (path-)connected sets. #m/def/topology 

Local (path) connectedness is neither weaker nor stronger than (path) connectedness,
however [[Locally path connected spaces have identical connected and path-connected components]].

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#state/tidy | #lang/en | #SemBr