|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
example of a semilocally simply connected space which is not locally simply connected
|
(Example)
|
|
|
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 $p\in HR$ 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 $X$ .
|
"example of a semilocally simply connected space which is not locally simply connected" is owned by antonio.
|
|
(view preamble | get metadata)
This object's parent.
- Attachments:
-
cone (Definition) by antonio
|
|
|
Cross-references: loops, contractible, cone, simply connected, neighborhood, represent, converge, circles, point, locally simply connected, semilocally simply connected, locally connected, connected, cone over, Hawaiian rings
This is version 2 of example of a semilocally simply connected space which is not locally simply connected, born on 2003-02-05, modified 2003-02-05.
Object id is 3972, canonical name is ExampleOfASemilocallySimplyConnectedSpaceWhichIsNotLocallySimplyConnected.
Accessed 2277 times total.
Classification:
| AMS MSC: | 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces ) | | | 57M10 (Manifolds and cell complexes :: Low-dimensional topology :: Covering spaces) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
