# example of a space that is not semilocally simply connected

An example of a space that is *not* semilocally simply connected is
the following: Let

$$HR=\bigcup _{n\in \mathbb{N}}\left\{(x,y)\in {\mathbb{R}}^{2}\right|{\left(x-\frac{1}{{2}^{n}}\right)}^{2}+{y}^{2}={\left(\frac{1}{{2}^{n}}\right)}^{2}\}$$ |

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 $HR$.

The Hawaiian rings

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.

Title | example of a space that is not semilocally simply connected |
---|---|

Canonical name | ExampleOfASpaceThatIsNotSemilocallySimplyConnected |

Date of creation | 2013-03-22 13:25:10 |

Last modified on | 2013-03-22 13:25:10 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 16 |

Author | mathcam (2727) |

Entry type | Example |

Classification | msc 57M10 |

Classification | msc 54D05 |

Defines | Hawaiian rings |

Defines | Hawaiian earrings |