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Closed set

  • Thread starter yifli
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  • #1
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Homework Statement


show that (the range of) a sequence of points in a metric space is in general not a closed set. Show that it may be a closed set.


2. The attempt at a solution
I don't know where to start.
For example, if we are given a sequence of real numbers and the distance between a and b is defined as |a-b|, it asks us to show that a sequence of real numbers is in general not a closed set?
 

Answers and Replies

  • #2
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Pick your favorite convergent sequence. Is this a closed set?
 
  • #3
HallsofIvy
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Note the difference between the sets {1, 1/2, 1/3,..., 1/n,...} and {0, 1, 1/2, 1/3, 1/4, ..., 1/n, ...}
 
  • #4
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Note the difference between the sets {1, 1/2, 1/3,..., 1/n,...} and {0, 1, 1/2, 1/3, 1/4, ..., 1/n, ...}
thank you. I can see {1, 1/2, 1/3,....} is not closed, but {0,1,1/2,1/3,...} is.
 

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