Linear ordered dense subset.

  • Thread starter cragar
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  • #1
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Homework Statement


If a linear ordered set P has a countable dense subset, then
[itex] |P| \leq 2^{\aleph_0} [/itex]

The Attempt at a Solution


because it has a linear order then all elements of P can be compared x<y .
And because it is dense that mean that I can find an element f
such that x<f<y for any x or y.
so we can partition this set into countable many pieces.
now either their are countably many elements between
any two elements. And the union of countably many things
with a countable number of objects is countable.
Or their are an uncountable number of things in between
any of the 2 dense elements, but this union would
be [itex] 2^{\aleph_0} [/itex] if anything larger was in between
then the cardinality would not work.
This is a little informal, just want to know if this is on the right track.
 

Answers and Replies

  • #2
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Homework Statement


If a linear ordered set P has a countable dense subset, then
[itex] |P| \leq 2^{\aleph_0} [/itex]

The Attempt at a Solution


because it has a linear order then all elements of P can be compared x<y .
And because it is dense that mean that I can find an element f
such that x<f<y for any x or y.
so we can partition this set into countable many pieces.

How will you define this partition??
 
  • #3
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3
Can i just take an arbitrary partition will some element w.
And because the set is linear ordered, their will be either 0 elements to the left or
at most [itex] 2^{\aleph_0} [/itex] to the left or right. I guess im assuming that.
does this look ok? Im thinking of using something like if I have a countable dense
subset on an interval I can only have [itex] 2^{\aleph_0} [/itex] limit points.
all thought I will probably need to prove that.
 
  • #4
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I don't really get it. Can you write it out more formally?
 
  • #5
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I guess I am going to pick a random element from the set call it x.
now I have partitioned the set into 2 sub sets.
I call them E and U. E is everything less than or equal to x and U is everything greater than x. Now either there is 0 elements in E or countable or 2^N,
and U must either have finite, countable or 2^N.
I was going to assume one of the partions have more than 2^N elements
and then say this was a contradiction. But I dont think that really proves it and I havn't
used the fact that the set has a countable dense subset.
Im not really sure what to do next.
Maybe I could assume there were more than 2^N elements
in between and 2 of my countable elements and then someohow get a contradiction by showing that
I couldn't embed a countable subset into a set with more than 2^N elements.
 
  • #6
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if I have a countable dense
subset I can only have [itex] 2^{\aleph_0} [/itex] limit points.
all thought I will probably need to prove that.

This is the direction you want to head in. Also, make sure to look up (or recall) the definition of the cardinal ##2^{\aleph_0}##. You wouldn't want to try to prove something that's true by definition.
 
  • #7
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so [itex] 2^{\aleph_0} [/itex] is the set of all possible combinations of the natural numbers. I still dojnt really know how to prove what I want.
I guess beacuse we have a countable dense subsets between any 2 of our
countable elements that their exists at most [itex] 2^{\aleph_0} [/itex]
limits points and there are only a countable many partitions in out set.
 
  • #8
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76
so [itex] 2^{\aleph_0} [/itex] is the set of all possible combinations of the natural numbers.

Yes. I would characterize ##2^{\aleph_0}## as the cardinality of the power set of the natural numbers (or, for that matter, any set of cardinality ##\aleph_0##). In general, ##2^{|X|}## is the cardinality of the power set of ##X##.

I still dojnt really know how to prove what I want.
I guess beacuse we have a countable dense subsets between any 2 of our
countable elements that their exists at most [itex] 2^{\aleph_0} [/itex]
limits points and there are only a countable many partitions in out set.

You have some good ideas (mixed in with some not-as-good ones), so let's try to get them organized. First let's give names to the pertinent sets to make things easier to read and write; let ##P## be the linearly ordered set and ##A## be the countable subset of ##P## which is dense in ##P##, and let ##\bar{A}## be the set of limit points of ##A##.

1) Forget about dense part for the moment. Can you show that ##|\bar{A}|\leq2^{\aleph_0}## using only the definition of limit point, the fact that ##|A|=\aleph_0##, and our definition for ##2^{\aleph_0}##? So your proof here should really depend only on the cardinality of ##A## and not at all on its relationship to the ambient space ##P##.

2) Now for the dense part ... Can you show that ##P=\bar{A}## using the definition of dense?

Out of curiosity, what definitions for limit point and dense are you using? Part 2 could potentially end up being very easy.
 
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