description/proof of Hausdorff maximal principle: chain of partially-ordered set is contained in maximal chain
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of chain of partially-ordered set.
- The reader knows a definition of set of maximal elements of partially-ordered set.
- The reader admits the proposition that any set with the ordering as containment is a partially-ordered set.
- The reader admits Zorn's lemma: for any set such that for each nonempty chain, the union of the chain is an element of the set, the set has a maximal element.
Target Context
- The reader will have a description and a proof of the Hausdorff maximal principle that any chain of any partially-ordered set is contained in a maximal chain.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(S\): \(\in \{\text{ the partially-ordered sets }\}\), with any partial ordering, \(\lt_S\)
\(C\): \(= \{\text{ the chains of } S\}\), with the partial ordering, \(\lt_C\), such that \(c_1 \lt_C c_2 \iff c_1 \subset c_2\)
\(c\): \(\in C\)
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Statements:
\(\exists c' \in \{\text{ the maximal elements of } C\} (c \subseteq c')\)
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2: Proof
Whole Strategy: Step 1: see that \(C\) is a partially-ordered set; Step 2: take the set of the chains of \(S\) that contain \(c\), \(C^` \subseteq C\), with the ordering as containment, \(\lt_{C^`}\), and apply Zorn's lemma: for any set such that for each nonempty chain, the union of the chain is an element of the set, the set has a maximal element to get a maximal element, \(c'\); Step 3: see that \(c'\) satisfies the conditions for this proposition.
Step 1:
\(C\) is indeed a partially-ordered set, by the proposition that any set with the ordering as containment is a partially-ordered set.
So, talking about the set of the maximal elements of \(C\) makes sense, although the set has not been known to be non-empty, yet.
Step 2:
Let us take the set of the chains of \(S\) that contain \(c\), \(C^` \subseteq C\), with the ordering as containment, \(\lt_{C^`}\), which is a partially-ordered set, as before.
For any nonempty chain of \(C^`\), \(\{c_j \vert j \in J\} \subseteq C^`\) where \(J\) is a possibly uncountable index set, \(\cup_{j \in J} c_j \in C^`\), because \(\cup_{j \in J} c_j \subseteq S\), because for each \(p \in \cup_{j \in J} c_j\), \(p \in c_j\) for a \(j \in J\), so, \(p \in c_j \subseteq S\); \(c \subseteq \cup_{j \in J} c_j\), because for each \(p \in c\), \(p \in c \subseteq c_j\) for each \(j \in J\), so, \(p \in \cup_{j \in J} c_j\); \(\cup_{j \in J} c_j\) is a chain of \(S\), because for each \(p_1, p_2 \in \cup_{j \in J} c_j\), when \(p_1, p_2 \in c_j\) for a \(j \in J\), \(p_1 \le p_2\) or \(p_2 \lt p_1\), because \(c_j\) is a chain of \(S\), and when \(p_1 \in c_j\) and \(p_2 \in c_l\) for some \(j \neq l\), \(c_j \subset c_l\) or \(c_l \subset c_j\), because \(\{c_j \vert j \in J\}\) is a chain of \(C^`\), so, \(p_1, p_2 \in c_l\) or \(p_1, p_2 \in c_j\), and \(p_1 \le p_2\) or \(p_2 \lt p_1\), because \(c_l\) and \(c_j\) are some chains of \(S\).
So, by Zorn's lemma: for any set such that for each nonempty chain, the union of the chain is an element of the set, the set has a maximal element, \(C^`\) has a maximal element, \(c' \in C^`\).
\(c' \in C^` \subseteq C\) is a maximal element of \(C\), because if there was a \(c'' \in C\) such that \(c' \subset c''\), \(c \subseteq c' \subset c''\), which would mean that \(c'' \in C^`\), a contradiction against that \(c'\) was maximal in \(C^`\).
So, \(c'\) is a one that satisfies the conditions for the proposition.
