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August 9th, 2015, 03:09 AM   #1
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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.
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August 9th, 2015, 06:34 AM   #2
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??? 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.
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August 9th, 2015, 07:52 AM   #3
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Yes, this is what I meant too.
Thanks
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September 8th, 2015, 04:52 PM   #4
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Quote:
Originally Posted by Country Boy View Post
??? 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.
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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