example of a semilocally simply connected space which is not locally simply connected

Let $H R$ be the Hawaiian rings, and define $X$ to be the cone over $H R .$ Then, $X$ is connected, locally connected, and semilocally simply connected, but not locally simply connected.

Too see this, let $p\\in HR$ be the point to which the circles converge in $H R,$ and represent $X$ as $HR\\times[0,1]/HR\\times\\left\\{0\\right\\}.$ Then, every small enough neighborhood of $q:=(p,1)\\in X$ fails to be simply connected. However, since $X$ is a cone, it is contractible, so all loops (in particular, loops in a neighborhood of $q$ ) can be contracted to a point within $X$ .

单词	ExampleOfASemilocallySimplyConnectedSpaceWhichIsNotLocallySimplyConnected
释义	example of a semilocally simply connected space which is not locally simply connected Let $H R$ be the Hawaiian rings, and define $X$ to be the cone over $H R .$ Then, $X$ is connected, locally connected, and semilocally simply connected, but not locally simply connected. Too see this, let $p\\in HR$ be the point to which the circles converge in $H R,$ and represent $X$ as $HR\\times[0,1]/HR\\times\\left\\{0\\right\\}.$ Then, every small enough neighborhood of $q:=(p,1)\\in X$ fails to be simply connected. However, since $X$ is a cone, it is contractible, so all loops (in particular, loops in a neighborhood of $q$ ) can be contracted to a point within $X$ .
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