Locally arcwise connected

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Let $(X,\tau)$ be a topological space and let $p \in X$. We say that $X$ is locally arcwise connected at $p$ if every neighborhood of $p$ contains an arcwise connected neighborhood of $p$. We say that $(X,\tau)$ is arcwise connected if it is arcwise connected at all points $p \in X$.

Properties

All open subsets of Peano continuum are locally arcwise connected