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August 9th, 2015, 03:09 AM  #1 
Newbie Joined: Aug 2015 From: Italy Posts: 5 Thanks: 0  Is there a topological space with fundamental group that does not contain anything?
Is there a topological space with fundamental group that does not contain anything? If G is a group $\displaystyle G \not = \emptyset$, and $\displaystyle e \in G$ where $\displaystyle e$ is identity element, so I think that does not exist. Is it correct? Thanks in advance. Last edited by skipjack; August 9th, 2015 at 08:01 AM. 
August 9th, 2015, 06:34 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
??? The "fundamental group" of a topological space is, first of all, a group! Every group has to contain at least an identity so the "fundamental group" must contain at least the identity.

August 9th, 2015, 07:52 AM  #3 
Newbie Joined: Aug 2015 From: Italy Posts: 5 Thanks: 0 
Yes, this is what I meant too. Thanks 
September 8th, 2015, 04:52 PM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  Added: The "fundamental group" of a topological space is the set of all homeomorphisms from the space to itself with composition of functions as operation. That always includes the identity map as its identity element.


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fundamental, group, space, topological 
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