TITLE: How many tacks fit in the plane?
QUESTION [17 upvotes]: Call a tack the one point union of three open intervals.  Can you fit an uncountable number of them on the plane?  Or is only a countable number?

REPLY [4 votes]: While the result of Young mentioned in the answer of Greg Kuperberg seems to be of purely topological interest, quite surprisingly, it has applications to study of the structure of the support of Sobolev functions: 
T. Bagby, P. M. Gauthier,
Note on the support of Sobolev functions.
Canad. Math. Bull. 41 (1998), no. 3, 257–260. 
(MathSciNet  review). 
The paper of Bagby and Gauthier contains a proof of Young's result (Lemma 2 in the paper). The authors knew the result of Moore, but they were not aware of the generalization obtained by Young.