I want a example of a set that it is uncountable and has measure zero and not compact? Cantor set has these properties except not compactness.
A set that it is uncountable, has measure zero, and is not compact
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1How about the union of the Cantor set and the set of all rational numbers? – 2014-08-23
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0An obvious solution is to note that any subset of the Cantor set has measure zero. There are $2^{\mathfrak c}$ subsets, and only $\mathfrak c$ of them are closed or countable, so $2^{\mathfrak c}$ subsets of the Cantor set serve as examples. – 2014-08-23
4 Answers
Take the union of Natural numbers with Cantor set which is an unbounded , uncountable set of Labesgue measure zero.
Just delete a point, say $0$, from the Cantor set and you'll get a set with the desired properties. In fact, since the Cantor set is perfect, any point will do.
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0I say not compact Your set is compact beacause cantor set is compact set – 2012-07-26
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2No, my set is not compact. Why do you think it is? – 2012-07-26
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0@kaveh: That is not true in general, $(0,1]$ is not compact, but we only removed $\{0\}$ from $[0,1]$. – 2012-07-26
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0While this answers the question as stated, I prefer the answer of @AsafKaragila, since it satisfies the much more interesting property that any neighbourhood of any point in his set contains a boundary point not in the set. So it is much farther from being closed. – 2012-07-26
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0yes I take it thanks – 2012-07-26
Recall the construction of the Cantor set. At each step we remove open intervals, now remove closed intervals.
The intersection is still non-empty, but the result is something homeomorphic to the irrational numbers, or more generally, Baire space.
This is not a compact set, since we can show that the points removed are in the closure of this new set, and thus it is not closed. However as a subset of the Cantor set it still has measure zero.
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0why The intersection is still non-empty, but the result is something homeomorphic to the irrational ? – 2012-07-26
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0We intersect open sets which contain compact sets, so the intersection is non-empty by the same argument as in the Cantor set. The result is not closed, that is the main point here. The fact this is homeomorphic to the irrationals is irrelevant. – 2012-07-26
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1You need also to argue that this set is not countable. However this is easy, since the difference between this set and the Cantor set is countable – it consists of the endpoints of the intervals removed. Corollary: The set is nonempty. – 2012-07-26
Let $B$ be the union of translates of the Cantor set by every integer $n$. Then $B$ is uncountable, has measure $0$. It is unbounded so not compact.
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0whats means one for every interval [n,n+1]? – 2012-07-26
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0I changed the wording to make it clearer. If $C$ is the Cantor set, let $B$ be the collection of all points of the form $n+c$, where $n$ ranges over the integers and $c$ ranges over $C$. – 2012-07-26
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0thanks I understand this explain – 2012-07-26