Description:
Definitions for the various versions of the Baire category theorem
Content:
In this note we give definitions
for the various versions of the Baire category theorem and summarize
the results from Brunner [1983c]
Definition: Let \((X,T)\) be a topological space.
- A set \(Y\subseteq X\) is nowhere dense if the closure
of \(Y\) has empty interior.
- A set \(Y\subseteq X\) is meager or of the first
category if \(Y\) is a countable union of nowhere dense sets.
- A set \(Y\subseteq X\) is perfect if \(Y\) is closed,
non-empty and has no isolated points.
- A set \(Y\subseteq X\) has the Baire property if
\((Y\setminus U)\cup(U\setminus Y)\) is meager for some open set \(U\).
- \((X,T)\) is regular if points are closed and every
neighborhood of a point contains a closed neighborhood of that point.
- A regular filter on \((X,T)\) is a subset \(F\) of
\(\cal P(X)\) which is closed under intersections and supersets such
that for some collection \(\cal U\) of open sets \(\cal U\) generates
\(F\) (That is, \(F = \{ y\subseteq X :\) for some finite subset
\(\cal U_0\) of \(\cal U\), \(\bigcap \cal U_0 \subseteq y\,\}\).)
and for some collection \(\cal C\) of closed sets \(\cal C\) generates
\(F\).
- \((X,T)\) is regular-closed if \((X,T)\) is regular and
any regular filter on \(X\) has a non-empty intersection.
- \((X,T)\) is Baire or is a Baire space
if the intersection of each countable sequence of dense, open sets
in \(X\) is dense in \(X\). (Equivalently, \((X,T)\)
is Baire if and only if \(X\) is not the union of a countable
sequence of nowhere dense sets.)
- \((X,T)\) is sequentially complete if every sequence
has a convergent subsequence.
- \((X,T)\) is Čech complete-I if it
satisfies
- \((X,T)\) is regular (every neighborhood of a point \(x\in S\)
contains a closed neighborhood of \(x\)) and
- there is a denumerable collection \(\{\cal C_n: n\in\omega\}\)
of open covers of \(X\) such that for any collection \(F\) of closed
subsets of \(X\) with the finite intersection property, if for each \(n\),
\(F\) contains a subset of diameter less than \(\cal C_n\) (that is, \(\exists u\in F)(\exists c\in\cal C_n)(u\subseteq
c)\)), then \(F\) has a non-empty intersection.
(This is the definition of Čech complete used in form [43 E].)
- \((X,T)\) is Čech complete-II if \((X,T)\) is
homeomorphic to a \(G_{\delta }\) set in a compact, \(T_2\) space.
(\(G_{\delta} \equiv\) countable intersection of open sets.)
(This is the definition of Čech complete used in \(K9\) below.
\(K9\) is equivalent to Form 106.)
- A collection \(\cal B\) of non-empty open sets is called a
regular pseudo-base for \((X,T)\) if:
- For each non-empty \(A\in T\), there is some \(B\in\cal B\)
such that \(cl B\subseteq A\) and
- If \(A\ne\emptyset\) is in \(T\) and \(A\subseteq B\) for
some \(B\in\cal B\), then \(A\in\cal B\).
- \((X,T)\) is pseudo-complete provided there is a
sequence \(\left(\cal B_n\right)_{n\in\omega}\) of regular pseudo-bases
such that for every regular filter \(F\) on \(X\), if \(F\) has a countable
base and meets each \(\cal B_n\) then \(F\) has non-empty intersection.
\((X,T)\) is co-compact if \(\exists\) a family \(F\) of
closed sets such that
- If \(G \subseteq F\) and \(G\) has the finite intersection
property, then \(\bigcap G \neq\emptyset\) and
- If \(x\in{\cal O}\), with \({\cal O}\) open, then
\(\exists A \in F\) with \(x \in A^{\circ}\) (the interior of \(A\))
and \(A \subseteq \) closure\(({\cal O}).\)
- \((X,T)\) is pseudo-compact if every continuous real
valued function on \((X,T)\) is bounded.
- \((X,T)\) is scattered or Boolean if it is
Hausdorff and has a basis of clopen sets.
Definition: Let \((X,d)\) be a metric space.
- \((X,d)\) is complete or Frechet complete if
Cauchy sequences converge.
- \((X,d)\) is Cantor complete if
\(\bigcap_{n\in\omega} A(n) \ne\emptyset\) for every sequence
\((A(n))_{n\in\omega}\) of non-empty closed sets satisfying
- \((\forall n\in\omega)(A(n+1)\subseteq A(n))\) and
- \(\lim_{n\to\infty}\hbox{diameter}(A(n)) = 0\).
- \((X,d)\) is Kuratowski complete if
\(\bigcap_{n\in\omega} A(n) \ne\emptyset\) for each nested sequence
\((A(n))_{n\in\omega}\) of non-empty closed sets for which
\(a(A(n)) \to 0\) where for \(A\subseteq X\),
\(a(A) = \inf\{ \max\{ \) diameter\((B) : B\in Y\} : Y\) is a finite
covering of \(A\}\).
- \((X,d)\) is filter complete if every Cauchy filter
converges.
| \(K1:\) |
Every Frechet Complete metric space is Baire. (This
is what is usually known as the Baire Category Theorem and is [43 F]
in the list of forms.) |
| \(K2:\) |
Every Cantor complete metric space is Baire. ([43 J]) |
| \(K3:\) |
Every metric Kuratowski complete space is Baire. |
| \(K4:\) |
Every sequentially compact metric space is Baire. |
| \(K5:\) |
Every countably compact (Every countable open coverhas a finite subcover.) metric space is Baire. |
| \(K6:\) |
Banach spaces (complete, normed, linear spaces) are Baire. |
| \(K7:\) |
Countably compact, regular spaces are Baire. |
| \(K8:\) |
Compact \(T_{2}\)-spaces are Baire. (This is Form 106) |
| \(K9:\) |
Čech complete-II spaces are Baire. |
| \(K10:\) |
Co-compact, \(T_{2}\) spaces are Baire. |
| \(K11:\) |
Separable, Frechet complete, metric spaces are Baire. |
| \(K12:\) |
Separable, countably compact, regular spaces are Baire. |
The following results are proved in \(ZF^{0}\):
\(K1\) and \(K2\) are equivalent to \(DC\). \(DC\) (Form 43) implies \(K9\) and
\(K10\). \(K1\) implies \(K6\). \(DMC\) (Dependent Multiple choice, Form [106 A])
implies \(K3\) implies \(K5\). \(K4\) for Cantor complete spaces implies \(DF=F\)
(Form 9). \(K11\) is provable in \(ZF^{0}\). \(DMC\) ([106 A]) implies \(K7\) implies \(K8\). \(DMC\) implies \(K9\) implies \(K8\). \(DMC\) implies \(K10\) implies
\(K8\). \(PW\) (Form 91) implies \(K12\).
The following independences are shown to hold in \(ZF^{0}\):
\(MC \not\rightarrow K4\), \(MC \not\rightarrow K6\),
BPI (Form 14) \(\not\rightarrow K8\) (Form 106), \(MC_{\hbox{LO}}\)
(Form 112) \(\not\rightarrow K7\), \(C(\aleph _{0},\infty ) \not
\rightarrow K8\) and none of the following are provable in \(ZF\)\(: Ki, i =
1, 2, 4, 6, 7, 8, 9, 10.\)
The following questions are asked:
- Are the following provable in \(ZF^{0}\)?: \(K3,\ K5,\ K12.\)
- Are \(Ki\), \(7 \le i \le 10,\) equivalent to \(DMC\) (Form [106 A]).
In Fossy/Morillion [1998]it is shown that \(K8\) implies
\(DMC\) from which it follows that \(Ki\), \(7\le i\le 10\) are all
equivalent to \(DMC\).
Howard-Rubin number:
28
Type:
Definitions
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