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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *locally connected* if the following holds.\nFor every point $x \\in X$ and every [[open set|02BH3MCBQB]] $U$ containing $x$, there is a [[connected|0SPE8WNWFV]] open set $V$ with $x \\in V \\subseteq U$.\n\n## Remarks\n\nExample: $\\R$ is locally connected.\nFor $x \\in U$ with $U$ open, some interval $(x - \\varepsilon, x + \\varepsilon)$ lies in $U$, and intervals are connected.\n\nNon-example: $\\Q$ with the subspace topology from $\\R$ is not locally connected.\nIts only nonempty connected subsets are single points, and single points are not open in $\\Q$.\n\nEquivalently, $X$ is locally connected if and only if every [[connected component|0JG9W3KA6B]] of every open subset of $X$ is open in $X$.\n",
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