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mathguy20
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Joined: 13 May 2006
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 Posted: Sat, 13 May 2006 08:55:54 UTC Post subject: Topology - Fundamental Group! |
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Greetings, first timer here. I've been working on this one problem but without much luck.
R_l is the real line with the lower limit topology.
Let  be the map  . Let  be the circle with the quotient topology induced by  . What is:  ?
My guess is that it is the trivial group but, I am not sure. Anyone willing to lend a hand? |
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qleak
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| Posted: Sat, 13 May 2006 17:45:21 UTC Post subject: |
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I guess one way to look at this is to eamine functions such as
 via 
if it is continuous then I suspect the fundamental group is the integers. If not then you are probably correct and the group is trivial (though this does not quite constitute a proof).
I think  is probably strictly finer than  so it is quite likely you are correct.
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mathguy20
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| Posted: Sat, 13 May 2006 18:03:40 UTC Post subject: |
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That's kind of what I was doing. I was saying that there's no continuous paths from I into  . But that was just going out on a limb, I didn't prove it. It would help if I knew what the space looked like. The only thing different that I can see is that  is now open. But how that helps, I am not sure. |
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Opalg
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| Posted: Sat, 13 May 2006 18:04:13 UTC Post subject: Re: Topology - Fundamental Group! |
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| mathguy20 wrote: | | My guess is that it is the trivial group but, I am not sure. Anyone willing to lend a hand? |
My guess is, you're right. In fact, is totally disconnected, so the the only loops in it are trivial. In other words, the only continuous functions from S^1 (with its usual topolgy) to S^1 (with the lower limit topology) are the constant functions. |
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mathguy20
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| Posted: Sat, 13 May 2006 18:14:34 UTC Post subject: Re: Topology - Fundamental Group! |
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| Opalg wrote: | | mathguy20 wrote: | | My guess is that it is the trivial group but, I am not sure. Anyone willing to lend a hand? |
My guess is, you're right. In fact, is totally disconnected, so the the only loops in it are trivial. In other words, the only continuous functions from S^1 (with its usual topolgy) to S^1 (with the lower limit topology) are the constant functions. |
That is very true. I have the fact that  is totally disconnected but I did not make the next jump. However, if the fundamental group of this space is the trivial group, then we have that is simply connected. That's what rubbed me the wrong way and caused me to doubt that the fundamental group is trivial. |
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Opalg
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| Posted: Sat, 13 May 2006 18:33:09 UTC Post subject: Re: Topology - Fundamental Group! |
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| mathguy20 wrote: | | However, if the fundamental group of this space is the trivial group, then we have that is simply connected. |
No. Part of the definition of a space being simply connected is that it should be path-connected.
I think that's why the statement of your problem mentioned the base point  . The fundamental group at any given base point is trivial, but you can't get (continuously) from one base point to another, so it would not make sense to talk about the fundamental group  (without mentioning a base point). |
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mathguy20
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Joined: 13 May 2006
Posts: 11
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| Posted: Sat, 13 May 2006 19:12:13 UTC Post subject: Re: Topology - Fundamental Group! |
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| Opalg wrote: | | mathguy20 wrote: | | However, if the fundamental group of this space is the trivial group, then we have that is simply connected. |
No. Part of the definition of a space being simply connected is that it should be path-connected.
I think that's why the statement of your problem mentioned the base point . The fundamental group at any given base point is trivial, but you can't get (continuously) from one base point to another, so it would not make sense to talk about the fundamental group (without mentioning a base point). |
Ah! you're right. I was confusing myself. Thank you both very much. |
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