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Infinite subset is dense

  • Thread starter Oxymoron
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  • #1
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My problem is:

Consider an infinite set [itex]X[/itex] with the finite complement topology. I want to show that any infinite subset [itex]A[/itex] of [itex]X[/itex] is dense in [itex]X[/itex].

Now, I can show that every point of [itex]X[/itex] is a limit point of [itex]A[/itex].

Can this help me in any way to show that [itex]A[/itex] is dense in [itex]X[/itex]. Or could someone provide me with an alternative method.

By the way, my understanding of dense is that a subset of a set is dense if its closure coincides with the set.
 

Answers and Replies

  • #2
Hurkyl
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Let me rephrase your question:

If every point of X is a limit point of A, then is A dense in X?
 
  • #3
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Yes, that is my question.
 
  • #4
Hurkyl
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So what do the definitions say?
 
  • #5
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Wait a sec, if every point of X is a limit point of A, then A is dense in X!
 
  • #6
matt grime
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SO, A is any infinite subset and you want to find the smallest closed set containing A. Since there are three kinds of closed set:

the empty set

a set containing a finite number of points

all of X


isn't the answer obvious? I mean, whichof those can contain A, given A is infinite?
 
  • #7
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isn't the answer obvious? I mean, whichof those can contain A, given A is infinite?
is it all of X?
 
  • #8
matt grime
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well, can you name a finite (or empty set) that contains an infinite subset?
 
  • #9
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nope, I can't.

So am I correct in thinking that by proving that every point of [itex]X[/itex] is the limit point of [itex]A[/itex], then [itex]A[/itex] is dense?
 
  • #10
Hurkyl
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Wait a sec, if every point of X is a limit point of A, then A is dense in X!
That is correct -- the point of my first post was to eliminate the unnecessary stuff, hoping you could see this when that's all that's left. :smile:

However, the approach matt has mentioned is a much easier way to do this problem... and is a fairly important theme to understand in general.
 
  • #11
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I agree, Matt's method was MUCH easier.
 
  • #12
HallsofIvy
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Once again- look at the DEFINITION of "dense"!
 

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