example of a semilocally simply connected space which is not locally simply connected
ExampleOfASemilocallySimplyConnectedSpaceWhichIsNotLocallySimplyConnected
2003-02-05 14:34:58
2003-02-05 14:36:33
Example
semilocally simply connected
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Let $HR$ be the Hawaiian rings, and define $X$ to be the cone over $HR.$ Then, $X$ is connected, locally connected, and semilocally simply connected, but {\em not} locally simply connected.
Too see this, let $p\in HR$ be the point to which the circles converge in $HR,$ and represent $X$ as $HR\cross [0,1]/ HR\cross\set{0}.$ 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 $X$.