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单词 SetsThatDoNotHaveAnInfimum
释义

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 havean infimum (nor a supremum). Intuitively this is clear, as the set isunboundedPlanetmathPlanetmath. 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 ) .

    Proof.

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

    If i<2, choose any j from the interval (i,2) (this is a real interval, but as the rational numbersPlanetmathPlanetmathare dense (http://planetmath.org/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 lowerbound 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 thereforejM2. Thus M2 contains an elementMathworldMathworld j smaller than i, which isa contradiction to the assumptionPlanetmathPlanetmath that i=inf(M2)

    Intuitively speaking, this example exploits the fact that doesnot have “enough elements”. More formally, as a metric spaceis not completePlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Complete). The M2 defined above is the real intervalM2:=(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 aninfimum as a subset of .

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

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

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

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更新时间:2025/5/4 16:27:57