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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *semilocally simply connected* if every point $x \\in X$ has an open neighbourhood $U$ with the following property.\nThe homomorphism $\\pi_1(U, x) \\to \\pi_1(X, x)$ of [[fundamental groups|S1XK3HXHFX]] induced by the inclusion $U \\hookrightarrow X$ is trivial.\nEquivalently, every loop in $U$ based at $x$ is null-homotopic in $X$.\n\n## Remarks\n\nExample: a manifold is semilocally simply connected.\nEach point has a neighbourhood $U$ homeomorphic to an open ball, so $\\pi_1(U, x)$ is already trivial.\n\nNon-example: the [[Hawaiian earring|73EBKQPP8Q]] $H = \\bigcup_{n \\ge 1} C_n$, where $C_n \\subset \\R^2$ is the circle of radius $1/n$ centred at $(1/n, 0)$, is not.\nEvery neighbourhood $U$ of the origin contains some $C_n$, and a loop around $C_n$ is not null-homotopic in $H$, because $H$ retracts onto $C_n$.\n\nThe cone on the Hawaiian earring is contractible, so it is semilocally simply connected, but it is not locally simply connected.\nThe spelling \"semi-locally simply connected\" is also common.\n",
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