example of a semilocally simply connected space which is not locally simply connected
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 not locally simply connected.
Too see this, let p∈HR be the point to which the circles converge in HR, and represent X as HR×[0,1]/HR×{0}. Then, every small enough neighborhood of q:= fails to be simply connected. However, since is a cone, it is contractible, so all loops (in particular, loops in a neighborhood of ) can be contracted to a point within .
Title | example of a semilocally simply connected space which is not locally simply connected |
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Canonical name | ExampleOfASemilocallySimplyConnectedSpaceWhichIsNotLocallySimplyConnected |
Date of creation | 2013-03-22 13:25:15 |
Last modified on | 2013-03-22 13:25:15 |
Owner | antonio (1116) |
Last modified by | antonio (1116) |
Numerical id | 5 |
Author | antonio (1116) |
Entry type | Example |
Classification | msc 54D05 |
Classification | msc 57M10 |