Hello, infinite-set reader. Set theory is back.
Starting this post, we’ve entered about Zorn’s Lemma. This discussion will start from several definitions and theorems that later lead to Zorn’s Lemma. Happy reading.
For any poset it is not guaranteed that every non-empty well ordered subset has an upper bound. It automatically does not have the smallest upper bound too. As an example . Any non-empty well ordered subset of the poset does not have the smallest upper bound.
In this discussion, we will discuss about posets that each of the non-empty well ordered subset have the smallest upper bound.
Given any poset with each of the non-empty well ordered subset having the smallest upper bound on . Given function with for all . For any defined -chain that denoted by under the condition :
(i) is well ordered subset with being the smallest element and is the largest element.
(ii) For all apply
(iii) The smallest upper bound of any non-empty subset of is in
In the above definition, the condition (ii), such terms clearly apply only to. It because, if is always true that . On the other hand, . Then there is a contradiction with the definition of -chain conditions (i) that is the largest element of . Next defined
Next will be given properties about -chain and which is presented in the theorems below.
Theorem : (more…)