sets that do not have an infimum
Some examples for sets that do not have an infimum:
A more interesting example: The set (again as a subset of ) .
Clearly, . Assume is an infimum of . Now we use the fact that is not rational, and therefore or .
If , choose any from the interval (this is a real interval, but as the rational numbers are dense (http://planetmath.org/Dense) in the real numbers, every nonempty interval in contains a rational number, hence such a exists).
Intuitively speaking, this example exploits the fact that does not have “enough elements”. More formally, as a metric space is not complete (http://planetmath.org/Complete). The defined above is the real interval intersected with . as a subset of does have an infimum (namely ), but as that is not an element of , does not have an infimum as a subset of .
This example also makes it clear that it is important to clearly state the superset one is working in when using the notion of infimum or supremum.
Of course all the ideas expressed here equally apply to the supremum, as the two notions are completely analogous (just reverse all inequalities).
|Title||sets that do not have an infimum|
|Date of creation||2013-03-22 13:09:51|
|Last modified on||2013-03-22 13:09:51|
|Last modified by||sleske (997)|