Description:
These are definitions from Brunner [1982b] and results
similar to the equivalence of [8 C] and
[8 D] to Form 8
and [10 H] to
Form 10. We also include some results from Brunner [1987b].
Content:
These are definitions from Brunner [1982b] and results
similar to the equivalence of [8 C] and
[8 D] to Form 8
and [10 H] to
Form 10. We also include some results from Brunner [1987b].
Definition:
- A topological space is \(\sigma\)-compact if it can be covered by a countable set of compact
sets.
- A space is weakly Lindelöf if every open cover
has a countable refinement. If \(C\) is an open cover of a space \(X\) a
refinement of \(C\) is an open cover \(C'\) of \(X\) such that for all
\(y\in C'\), there is a \(z\) in \(C\) such that \(y \subseteq z\).
- A space \(X\) is paracompact if \(X\) is Hausdorff and every
open covering of \(X\) has an open, locally finite refinement. (An open
covering \(C\) of \(X\) is locally finite if for all \(x\in X\), there is
an open set containing \(x\) that intersects only finitely many elements
of \(C\). In general, a family of subsets \(S\) of \(X\) is called locally
finite if each point in \(X\) has a neighborhood meeting a finite number
of sets in \(S\).)
- An ultra metric group is a topological group with a
metrizable topology using a metric \(d\) that satisfies for all \(x,\ y\),
and \(z\) in the group,
\[d(x,y)\le \max (d(x,z),d(y,z)).\]
The results from Brunner [1982b] are:
- It is not possible to prove, without \(AC\), that every weakly
Lindelöf space is Lindelöf since in the basic Cohen model \(\cal M1\),
\({\Bbb R}\) is weakly Lindelöf but not Lindelöf.
- The assertion that every \(\sigma\)-compact, locally compact
space with countable neighborhood bases is Lindelöf implies Form 32
(\(C(\aleph _{0},\aleph _{0})\)).
- In the basic Cohen model, \(\cal M1\), \(C(\aleph_0,\aleph_0)\)
holds and there is a \(\sigma\)-compact locally compact space with
countable neighborhood bases which is not Lindelöf.
- \(C(\aleph _{0},<\aleph _{0})\) is implied by "Every \(\sigma
\)-compact, ultra metric group without isolated points is Lindelöf."
- The assertion "Every \(\sigma \)-compact, regular space without
isolated points is Lindelöf" implies Form 9 (Dedekind finite = Finite).
The results from Brunner [1987b] are:
- In \(\cal N3\), \(L\) is a paracompact and weakly Lindelöf
topological space which is not Lindelöf. (In \(\cal N3\), \(L=(A\times\Bbb
Z)\cup \Bbb Z\), where the ordering on \(L\) is defined as follows:
for \(a, b\in A\) and \(m, n\in\Bbb Z\), \((a,n) < (b,n)\) if \(a < b\) and
(\(a,n) < m < (b,m)\) if \(n < m\). The topology on \(L\) is the order
topology.)
- In \(\cal M1\), every \(T_1\) Lindelöf space is compact.
- In \(\cal M17\), every countably compact space is compact.
Howard-Rubin number:
43
Type:
Definitions and summaries
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