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November 14

Lower sets

Do the lower sets in the real numbers consist precisely of the empty set, the set of all real numbers, the set of real numbers less than some fixed real number r, and the set of real numbers less than or equal to some fixed real number? GeoffreyT2000 (talk) 19:50, 14 November 2015 (UTC)[reply]

I am not really familiar with the use of the term lower set for this notion (I'd have called it an "initial segment"), but based on the notion described in the link (a set that is closed downwards in the ordering), the answer is yes. This is essentially the least upper bound property for the reals. Note that if you substitute "rational" for "real", it is no longer true. --Trovatore (talk) 19:58, 14 November 2015 (UTC)[reply]