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38-532: [REDACTED] Look up Hausdorff in Wiktionary, the free dictionary. Hausdorff may refer to: People [ edit ] Felix Hausdorff (1868–1942), German mathematician after whom Hausdorff spaces are named Natasha Hausdorff (born 1989), British barrister , international news commentator, and Israel advocate Other [ edit ] A Hausdorff space , when used as an adjective, as in "the real line

76-457: A totally ordered subset ) is contained in a maximal chain C {\displaystyle C} (i.e., a chain that is not contained in a strictly larger chain in P {\displaystyle P} ). In general, there may be several maximal chains containing a given chain. An equivalent form of the Hausdorff maximal principle is that in every partially ordered set, there exists

114-465: A maximal chain C {\displaystyle C} . By the hypothesis of Zorn's lemma, C {\displaystyle C} has an upper bound x {\displaystyle x} in P {\displaystyle P} . If y ≥ x {\displaystyle y\geq x} , then C ~ = C ∪ { y } {\displaystyle {\widetilde {C}}=C\cup \{y\}}

152-553: A maximal chain. (Note if the set is empty, the empty subset is a maximal chain.) This form follows from the original form since the empty set is a chain. Conversely, to deduce the original form from this form, consider the set P ′ {\displaystyle P'} of all chains in P {\displaystyle P} containing a given chain C 0 {\displaystyle C_{0}} in P {\displaystyle P} . Then P ′ {\displaystyle P'}

190-740: A partially ordered set, a totally ordered subset is also called a chain. Thus, the maximal principle says every chain in the set extends to a maximal chain. The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over ZF ( Zermelo–Fraenkel set theory without the axiom of choice). The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33). The Hausdorff maximal principle states that, in any partially ordered set P {\displaystyle P} , every chain C 0 {\displaystyle C_{0}} (i.e.,

228-432: A subset T ⊂ F {\displaystyle T\subset F} a tower (over C 0 {\displaystyle C_{0}} ) if There exists at least one tower; indeed, the set of all sets in F {\displaystyle F} containing C 0 {\displaystyle C_{0}} is a tower. Let T 0 {\displaystyle T_{0}} be

266-445: A union of a totally ordered set of chains is a chain. Since C {\displaystyle C} contains C 0 {\displaystyle C_{0}} , it is an element of P ′ {\displaystyle P'} . Also, since any chain containing C {\displaystyle C} is contained in C {\displaystyle C} as C {\displaystyle C}

304-462: Is Hausdorff" Hausdorff dimension , a measure theoretic concept of dimension Hausdorff distance or Hausdorff metric, which measures how far two compact non-empty subsets of a metric space are from each other Hausdorff density Hausdorff maximal principle Hausdorff measure Hausdorff moment problem Hausdorff paradox Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

342-462: Is Hausdorff" Hausdorff dimension , a measure theoretic concept of dimension Hausdorff distance or Hausdorff metric, which measures how far two compact non-empty subsets of a metric space are from each other Hausdorff density Hausdorff maximal principle Hausdorff measure Hausdorff moment problem Hausdorff paradox Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

380-442: Is a chain containing C {\displaystyle C} and so by maximality, C ~ = C {\displaystyle {\widetilde {C}}=C} ; i.e., y ∈ C {\displaystyle y\in C} and so y = x {\displaystyle y=x} . ◻ {\displaystyle \square } If A

418-412: Is a chain in P {\displaystyle P} , then the union of Q {\displaystyle Q} is also orthonormal by the same argument as above and so is an upper bound of Q {\displaystyle Q} . Thus, by Zorn's lemma, P {\displaystyle P} contains a maximal element A {\displaystyle A} . (So, the difference

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456-428: Is a chain strictly larger than Q {\displaystyle Q} , a contradiction. ◻ {\displaystyle \square } For the purpose of comparison, here is a proof of the same fact by Zorn's lemma. As above, let P {\displaystyle P} be the set of all orthonormal subsets of H {\displaystyle H} . If Q {\displaystyle Q}

494-429: Is a chain, the hypothesis of Zorn's lemma (every chain has an upper bound) is satisfied for P ′ {\displaystyle P'} and thus P ′ {\displaystyle P'} contains a maximal element or a maximal chain in P {\displaystyle P} . Conversely, if the maximal principle holds, then P {\displaystyle P} contains

532-453: Is a maximal element if and only if C ~ = C {\displaystyle {\widetilde {C}}=C} . Thus, we are done if we can find a C {\displaystyle C} such that C ~ = C {\displaystyle {\widetilde {C}}=C} . Fix a C 0 {\displaystyle C_{0}} in F {\displaystyle F} . We call

570-618: Is a maximal orthonormal subset. First, if S , T {\displaystyle S,T} are in Q {\displaystyle Q} , then either S ⊂ T {\displaystyle S\subset T} or T ⊂ S {\displaystyle T\subset S} . That is, any given two distinct elements in A {\displaystyle A} are contained in some S {\displaystyle S} in Q {\displaystyle Q} and so they are orthogonal to each other (and of course, A {\displaystyle A}

608-466: Is a subset of the unit sphere in H {\displaystyle H} ). Second, if B ⊋ A {\displaystyle B\supsetneq A} for some B {\displaystyle B} in P {\displaystyle P} , then B {\displaystyle B} cannot be in Q {\displaystyle Q} and so Q ∪ { B } {\displaystyle Q\cup \{B\}}

646-422: Is a union, C {\displaystyle C} is in fact a maximal element of P ′ {\displaystyle P'} ; i.e., a maximal chain in P {\displaystyle P} . The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is somehow similar to this proof. Indeed, first assume Zorn's lemma. Since a union of a totally ordered set of chains

684-418: Is any collection of sets, the relation "is a proper subset of" is a strict partial order on A . Suppose that A is the collection of all circular regions (interiors of circles) in the plane. One maximal totally ordered sub-collection of A consists of all circular regions with centers at the origin. Another maximal totally ordered sub-collection consists of all circular regions bounded by circles tangent from

722-414: Is comparable in T 0 {\displaystyle T_{0}} , either A ~ ⊂ C {\displaystyle {\widetilde {A}}\subset C} or C ⊂ A ~ {\displaystyle C\subset {\widetilde {A}}} . In the first case, A ~ {\displaystyle {\widetilde {A}}}

760-598: Is comparable in T 0 {\displaystyle T_{0}} ; i.e., is in Γ {\displaystyle \Gamma } . This completes the proof of the claim that Γ {\displaystyle \Gamma } is a tower. Finally, since Γ {\displaystyle \Gamma } is a tower contained in T 0 {\displaystyle T_{0}} , we have T 0 = Γ {\displaystyle T_{0}=\Gamma } , which means T 0 {\displaystyle T_{0}}

798-488: Is in U {\displaystyle U} . Hence, U {\displaystyle U} is a tower. Now, since U ⊂ T 0 {\displaystyle U\subset T_{0}} and T 0 {\displaystyle T_{0}} is the intersection of all towers, U = T 0 {\displaystyle U=T_{0}} , which implies C ~ {\displaystyle {\widetilde {C}}}

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836-408: Is in U {\displaystyle U} . In the second case, we have A ⊂ C ⊂ A ~ {\displaystyle A\subset C\subset {\widetilde {A}}} , which implies either A = C {\displaystyle A=C} or C = A ~ {\displaystyle C={\widetilde {A}}} . (This

874-442: Is partially ordered by set inclusion. Thus, by the maximal principle in the above form, P ′ {\displaystyle P'} contains a maximal chain C ′ {\displaystyle C'} . Let C {\displaystyle C} be the union of C ′ {\displaystyle C'} , which is a chain in P {\displaystyle P} since

912-438: Is that the maximal principle gives a maximal chain while Zorn's lemma gives a maximal element directly.) The idea of the proof is essentially due to Zermelo and is to prove the following weak form of Zorn's lemma , from the axiom of choice . (Zorn's lemma itself also follows from this weak form.) The maximal principle follows from the above since the set of all chains in P {\displaystyle P} satisfies

950-508: Is the moment we needed to collapse a set to an element by the axiom of choice to define A ~ {\displaystyle {\widetilde {A}}} .) Either way, we have A ~ {\displaystyle {\widetilde {A}}} is in U {\displaystyle U} . Similarly, if C ⊂ A {\displaystyle C\subset A} , we see A ~ {\displaystyle {\widetilde {A}}}

988-417: Is the union of T 0 {\displaystyle T_{0}} , C ~ ⊂ C {\displaystyle {\widetilde {C}}\subset C} and thus C ~ = C {\displaystyle {\widetilde {C}}=C} . ◻ {\displaystyle \square } The Bourbaki–Witt theorem can also be used to prove

1026-489: Is totally ordered. Let C {\displaystyle C} be the union of T 0 {\displaystyle T_{0}} . By 2., C {\displaystyle C} is in T 0 {\displaystyle T_{0}} and then by 3., C ~ {\displaystyle {\widetilde {C}}} is in T 0 {\displaystyle T_{0}} . Since C {\displaystyle C}

1064-736: The above conditions. By the axiom of choice, we have a function f : P ( P ) − { ∅ } → P {\displaystyle f:{\mathfrak {P}}(P)-\{\emptyset \}\to P} such that f ( S ) ∈ S {\displaystyle f(S)\in S} for the power set P ( P ) {\displaystyle {\mathfrak {P}}(P)} of P {\displaystyle P} . For each C ∈ F {\displaystyle C\in F} , let C ∗ {\displaystyle C^{*}} be

1102-463: The 💕 [REDACTED] Look up Hausdorff in Wiktionary, the free dictionary. Hausdorff may refer to: People [ edit ] Felix Hausdorff (1868–1942), German mathematician after whom Hausdorff spaces are named Natasha Hausdorff (born 1989), British barrister , international news commentator, and Israel advocate Other [ edit ] A Hausdorff space , when used as an adjective, as in "the real line

1140-649: The intersection of all towers, which is again a tower. Now, we shall show T 0 {\displaystyle T_{0}} is totally ordered. We say a set C {\displaystyle C} is comparable in T 0 {\displaystyle T_{0}} if for each A {\displaystyle A} in T 0 {\displaystyle T_{0}} , either A ⊂ C {\displaystyle A\subset C} or C ⊂ A {\displaystyle C\subset A} . Let Γ {\displaystyle \Gamma } be

1178-443: The right to the y-axis at the origin. If (x 0 , y 0 ) and (x 1 , y 1 ) are two points of the plane R 2 {\displaystyle \mathbb {R} ^{2}} , define (x 0 , y 0 ) < (x 1 , y 1 ) if y 0 = y 1 and x 0 < x 1 . This is a partial ordering of R 2 {\displaystyle \mathbb {R} ^{2}} under which two points are comparable only if they lie on

Hausdorff - Misplaced Pages Continue

1216-610: The same horizontal line. The maximal totally ordered sets are horizontal lines in R 2 {\displaystyle \mathbb {R} ^{2}} . By the Hausdorff maximal principle, we can show every Hilbert space H {\displaystyle H} contains a maximal orthonormal subset A {\displaystyle A} as follows. (This fact can be stated as saying that H ≃ ℓ 2 ( A ) {\displaystyle H\simeq \ell ^{2}(A)} as Hilbert spaces.) Let P {\displaystyle P} be

1254-700: The set of all A {\displaystyle A} in T 0 {\displaystyle T_{0}} such that either A ⊂ C {\displaystyle A\subset C} or C ~ ⊂ A {\displaystyle {\widetilde {C}}\subset A} . We claim U {\displaystyle U} is a tower. The conditions 1. and 2. are again straightforward to check. For 3., let A {\displaystyle A} be in U {\displaystyle U} . If A ⊂ C {\displaystyle A\subset C} , then since C {\displaystyle C}

1292-568: The set of all x ∈ P − C {\displaystyle x\in P-C} such that C ∪ { x } {\displaystyle C\cup \{x\}} is in F {\displaystyle F} . If C ∗ = ∅ {\displaystyle C^{*}=\emptyset } , then let C ~ = C {\displaystyle {\widetilde {C}}=C} . Otherwise, let Note C {\displaystyle C}

1330-437: The set of all orthonormal subsets of the given Hilbert space H {\displaystyle H} , which is partially ordered by set inclusion. It is nonempty as it contains the empty set and thus by the maximal principle, it contains a maximal chain Q {\displaystyle Q} . Let A {\displaystyle A} be the union of Q {\displaystyle Q} . We shall show it

1368-500: The set of all sets in T 0 {\displaystyle T_{0}} that are comparable in T 0 {\displaystyle T_{0}} . We claim Γ {\displaystyle \Gamma } is a tower. The conditions 1. and 2. are straightforward to check. For 3., let C {\displaystyle C} in Γ {\displaystyle \Gamma } be given and then let U {\displaystyle U} be

1406-515: The title Hausdorff . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Hausdorff&oldid=1251224907 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Hausdorff From Misplaced Pages,

1444-519: The title Hausdorff . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Hausdorff&oldid=1251224907 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Hausdorff maximal principle In

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