In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set.
The requirement that a set be non-empty is frequently found in mathematical hypotheses. One reason is that we can easily make logical errors when we make hypotheses about the empty set, which is often nonintuitive and can be tricky to reason about correctly (see Empty set for further discussion of this). Thus, this hypothesis of nonemptiness can often be removed under a more careful treatment. On the other hand, there certainly are times when the empty set is a special case and really does need to be excluded from a hypothesis.
A common example where both of these situations arise is the axiom of choice. Although this axiom can be stated in several ways, for each standard way of stating it, there are two places where the term "nonempty" could be used. Often you will find the term used in both places, whereas in fact it is needed in only one. (See Axiom of choice for further discussion of this example).
In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set.
One reason is that we can easily make logical errors when we make hypotheses about the empty set, which is often nonintuitive and can be tricky to reason about correctly (see Empty set for further discussion of this).
A common example where both of these situations arise is the axiom of choice.
We will begin with a closed interval, and then, imitating the construction of Cantor set, we will inductively delete each rational number in it together with an open interval.
is clearly nonempty, does not contain any rational number, and also it is compact, being a countable intersection of decreasing compact sets.
This is version 10 of a nonempty perfect subset of
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