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Question about open sets in (infinite,5] 
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#1
Apr412, 05:49 AM

P: 688

The stupid question of the day.
If S is the real interval (infinite, 5], and I can find a metric d so that (S,d) is a metric space, then, is, for example, (4, 5] an open set in (S,d) ? I say this because, the way I'm reading the definition of an open ball, the open ball B(5,1) is the interval (4,5] and not the interval (4,6), since the points in (5,6) do not belong to the metric space (S,d). So every open ball in (4,5] centered in 5 is completely contained in (4,5]. 


#3
Apr412, 06:55 AM

P: 688

Thanks, micromass, just checking the fundamentals.
By the way, I apologize for the phrasing of the question; if the metric d is not specifically defined, then there is really no way to tell. As someone else pointed me out, the question should have referred to the restriction of the Euclidean metric to the set S. 


#4
Apr512, 06:38 PM

P: 117

Question about open sets in (infinite,5]
In that case you automatically have to deal with the quotient topology on S and obviously (4,5] is certainly the intersection of an open set of R and S. Of course just using the definition of metric also works. All points with distance smaller then 1 are in B(5,1) but, this means of course all point that are in your space. Otherwise it wouldn't make much sence.



#5
Apr1212, 06:50 PM

P: 9

If the question were specifically about the regular Euclidean metric, then the answer is yes. 


#6
Apr1312, 01:02 AM

P: 117

... read third post



#7
Apr1312, 06:51 AM

P: 688

Thanks all for your answers! My actual doubt was about open balls with the Euclidean metric, but I did a awful job formulating it  it's clear now.



#8
Apr1312, 08:55 PM

P: 312




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