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How many tacks fit in the plane?

define letter T as subset of R^2 , consisting of two line intervals AB and CD. C /in AB and AB -| CD. AB, CD, AC, BC > 0 . WHat is the maximal cardinality of some disjoint union of some T's on R^2 ?

I was thinking about that one , that could be reduced to the space partitioning problem , and then counting partitions of 2d space. The previous is a problem, as it is not so easy to get a convex hull for arbitrary point P, cause we can always construct sequence S of letters T's of which the "major points" ( say A , B, C or D ) would be convergent to the P, thus not allowing on convex hull construction.

Second idea is to assign a triangle ABD to each T. But again such triangle can be wholly covered by some infinite sequence of similar triangles, which union is convergent to triangle ABD.

If no solution is given, can I ask at least for some tip ? regards :D

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    $\begingroup$ Possible duplicate question: mathoverflow.net/questions/27244/… $\endgroup$
    – Joel David Hamkins
    Jan 29, 2012 at 12:29
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    $\begingroup$ Every uncountable closed set has cardinality of $2^{\aleph_0}$, this includes non-degenerate intervals, T's and of course $\mathbb R^2$. This should tell you that every union would have the same cardinality as $\mathbb R^2=2^{\aleph_0}$. Of course if you are asking on the possible cardinality of partitioning the plane into T's then I apologize for this long comment. :-) $\endgroup$
    – Asaf Karagila ♦
    Jan 29, 2012 at 12:31
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    $\begingroup$ Asaf, I believe the point the OP is after is the fact that one can have at most countably many disjoint T's in the plane. (There is no partition of the plane into T's.) $\endgroup$
    – Joel David Hamkins
    Jan 29, 2012 at 12:34
  • $\begingroup$ Pardon me, but I at all don't get the first reasoning of – Asaf Karagila 4 mins ago, as I don't see whether problem's set is countable or uncountable. :D $\endgroup$
    – paul424
    Jan 29, 2012 at 13:00
  • $\begingroup$ Paul, you can only have countably many disjoint T's in the plane, for the reasons explained in the answers to the duplicate question. (I believe that Asaf took your quesiton literally, and is telling you that the union of the set of T's has size continuum, just because each T already has size continuum. But I believe you are asking about the number of T's, not the size of the union of these T's.) $\endgroup$
    – Joel David Hamkins
    Jan 29, 2012 at 15:01

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