example of a space that is not semilocally simply connected
ExampleOfASpaceThatIsNotSemilocallySimplyConnected
2003-02-04 18:20:59
2006-06-06 01:56:29
Example
semilocally simply connected
Hawaiian rings
Hawaiian earrings
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An example of a space that is \emph{not} semilocally simply connected is
the following: Let
$$HR=\bigcup_{n\in\BN}\left\{(x,y)\in \BR^2\,\bigg\vert\,\left(x-\frac{1}{2^n}\right)^2+y^2=
\left(\frac{1}{2^n}\right)^2\right\}$$
endowed with the subspace topology. Then $(0,0)$ has no simply connected
neighborhood. Indeed every neighborhood of $(0,0)$ contains (ever diminshing)
homotopically non-trivial loops. Furthermore these loops are homotopically non-trivial even when considered as loops in $HR$.
\begin{figure*}[htbp]
\centering
\begin{pspicture}(0,-2)(4,3)
\pscircle(2,0){2}
\pscircle(1,0){1}
\pscircle(.5,0){.5}
\pscircle(.25,0){.25}
\pscircle(.125,0){.125}
\pscircle(.625,0){.625}
\pscircle(0.015625,0){0.015625}
\end{pspicture}
\caption{The Hawaiian rings}
\end{figure*}
It is essential in this example that $HR$ is endowed with the topology
induced by its inclusion in the plane. In contrast, the same set endowed with
the CW topology is just a bouquet of countably many circles and (as any CW
complex) it is semilocaly simply connected.