Partitioning the Interval

Geometry Level 4

Can [ 0 , 1 ] [0,1] be expressed as the countable disjoint union of at least 2 closed sets?

This is related to the 1-D analogue of this problem .

It can clearly be written as the disjoint union of 1 closed set, namely [ 0 , 1 ] [ 0,1 ] .
It can clearly be written as the uncountable disjoint union of closed sets, namely place every point in it's own set.

Warning: I do not know of any "elementary" proof of this theorem.

No Yes

Calvin Lin by Calvin Lin

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1 solution

Mark Hennings
Mar 9, 2017

This is a special case of a result by Sierpinski .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Yes, that's the general theorem. Do you think it's worthwhile to figure out the proof for this specific case? It does seem like we need all of the properties mentioned, and I can't find and special simplification of [ 0 , 1 ] [0,1] that would allow for a simpler, alternative approach.

Calvin Lin Staff - 4 years, 3 months ago

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Some of the comments on the MSE page I reference claim to have broken the [ 0 , 1 ] [0,1] problem down to a level which does not require the BCT; I have not had the time to check it out...

Mark Hennings - 4 years, 3 months ago

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