Description:
Results from Brunner [1982a]
Content:
In Brunner [1982a] is a study, in \(ZF^0\) without \(AC\), of conditions under which a topological space is well orderable. These conditions are summarized below.
Theorem: (2.4 from Brunner [1982a]) Assume \(X\) is a \(T_{2}\) topological space
- Assuming Form 163, \(X\) is well orderable if (a) \(X\) is hereditarily separable or (b) \(X\) is hereditarily Lindelöf and linearly orderable.
- Assuming Form 164, \(X\) is well orderable if (a) \(X\) is hereditarily Lindelöf or (b) \(X\) is first countable and Lindelöf or (c) \(X\) is first countable and hereditarily Souslin or (d) \(X\) is second countable or (e) \(X\) is separable.
- Assuming Form 133, \(X\) is well orderable if (a) \(X\) is hereditarily Souslin or (b) \(X\) is linearly orderable. (Note: \(X\) is linearly orderable means that the topology on \(X\) is the order topology of some linear order, \(X\) is well orderable means that the set \(X\) is well orderable but the topology need not be the order topology of the well ordering.)
Definition:
- \(X\) is first countable if every point has a countable neighborhood base.
- \(X\) is second countable is \(X\) has a countable basis.
- \(X\) is Souslin if every family of disjoint open sets is countable.
- \(X\) is metacompact if every open cover has a point finite refinement. (See Form 135 for a definition of point finite.)
Corollary: (2.5 in Brunner [1982a]) Assuming Form 164,the following are equivalent for a metric space \(X\): (1) \(X\) is separable,(2) \(X\) is second countable, (3) \(X\) is Souslin and (4) \(X\) is Lindelöf.
Theorem: (2.6) Under the assumption of Form 133 and Form 17 (Ramsey's Theorem) a regular, Lindelöf, Souslin and hereditarily metacompact space is well orderable.
Theorem: (2.7) Assuming Form 133, a \(T_2\) topologicalspace \(X\) is well orderable if \(X^2\) is hereditarily normal and \(X\) is Lindelöf.
Theorem (2.8) Assuming Form 164, a \(T_2\) space is well orderable if \(X^2\) is hereditarily locally Lindelöf and \(X\) is hereditarily Souslin.
Finally some counter examples:
- In the model \(\cal M1\) (The basic Cohen model) \(X = [0,1]\) (the closed interval in \({\Bbb R})\) with the order topology is compact, metric,separable, second countable, hereditarily Souslin and linearly ordered, but \(X\) is not well orderable.
- In the model \(\cal N3\) (The ordered Mostowski model of \(ZF^0\)): Let \(X =[a,b]\) be an interval in the set of atoms with the order topology. \(X\) is compact, hereditarily normal (that is \(T_5\)) and \(T_2\). Further, each finite power of \(X\) is compact, \(T_4\) and hereditarily Souslin, but \(X\) is not well orderable.
- In the model \(\cal N1\) (The basic Fraenkel model with a denumerable set of atoms, using the group of all permutations of the atoms and finite supports): Let \(X = [A]^{<\omega }\) (all finite subsets of the set of atoms) with a basis for the topology all set of the form \(\{x: E\subseteq x\)and \(x\cap F = \emptyset\}\) where \(E\) and \(F\) range over the finite subsets of \(U\). The space \(X\) is Souslin, perfectly regular and not well orderable.
Howard-Rubin number:
42
Type:
Summary of results
Back