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open set in lower limit topology

 
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tps_report
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PostPosted: Tue, 17 Oct 2006 20:47:50 UTC    Post subject: open set in lower limit topology Reply with quote
I'm looking at the id map from to . I was told this function is continous since the inverse image of any open set in is open in . This is where I'm confused.

open sets in are of the form (a,b) and open sets in are of the form [a,b). I'm I just missing something clear or why is (a,b) open in
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Matt
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PostPosted: Tue, 17 Oct 2006 21:23:08 UTC    Post subject: Re: open set in lower limit topology Reply with quote
tps_report wrote:
why is (a,b) open in

It is because you can surround any point with a half open interval that is also contained in (a,b).
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Sam Snyder
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PostPosted: Fri, 10 Oct 2008 15:42:11 UTC    Post subject: Re: open set in lower limit topology Reply with quote
Matt wrote:
tps_report wrote:
why is (a,b) open in

It is because you can surround any point with a half open interval that is also contained in (a,b).


The reason that the interval (a,b) is open in is that intervals of the form [a,b) form a Basis Set for . This means that the Lower Limit Topology is made up of all possible unions of intervals of the form [a,b).

In particular:

This is why the interval (a,b) is open in the Lower Limit Topology, .
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