sets that do not have an infimum

Some examples for sets that do not have an infimumMathworldPlanetmath:

  • The set M1:= (as a subset of ) does not have an infimum (nor a supremum). Intuitively this is clear, as the set is unboundedPlanetmathPlanetmath. The (easy) formal proof is left as an exercise for the reader.

  • A more interesting example: The set M2:={x:x22,x>0} (again as a subset of ) .


    Clearly, inf(M2)>0. Assume i>0 is an infimum of M2. Now we use the fact that 2 is not rational, and therefore i<2 or i>2.

    If i<2, choose any j from the interval (i,2) (this is a real interval, but as the rational numbersPlanetmathPlanetmath are dense ( in the real numbers, every nonempty interval in contains a rational number, hence such a j exists).

    Then j>i, but j<2, hence j2<2 and therefore j is a lower bound for M2, which is a contradictionMathworldPlanetmathPlanetmath.

    On the other hand, if i>2, the argumentMathworldPlanetmath is very similar: Choose any j from the interval (2,i). Then j<i, but j>2, hence j2>2 and therefore jM2. Thus M2 contains an elementMathworldMathworld j smaller than i, which is a contradiction to the assumptionPlanetmathPlanetmath that i=inf(M2)

    Intuitively speaking, this example exploits the fact that does not have “enough elements”. More formally, as a metric space is not completePlanetmathPlanetmathPlanetmathPlanetmath ( The M2 defined above is the real interval M2:=(2,) intersected with . M2 as a subset of does have an infimum (namely 2), but as that is not an element of , M2 does not have an infimum as a subset of .

    This example also makes it clear that it is important to clearly state the supersetMathworldPlanetmath one is working in when using the notion of infimum or supremum.

    It also illustrates that the infimum is a natural generalizationPlanetmathPlanetmath of the minimum of a set, as a set that does not have a minimum may still have an infimum (such as M2).

    Of course all the ideas expressed here equally apply to the supremum, as the two notions are completely analogous (just reverse all inequalitiesMathworldPlanetmath).

Title sets that do not have an infimum
Canonical name SetsThatDoNotHaveAnInfimum
Date of creation 2013-03-22 13:09:51
Last modified on 2013-03-22 13:09:51
Owner sleske (997)
Last modified by sleske (997)
Numerical id 9
Author sleske (997)
Entry type Example
Classification msc 06A06
Related topic InfimumAndSupremumForRealNumbers