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\centerline{\bf Forms [14 CM] and [43 W] through [43 AC]}
\medskip
\item{}{\bf [14 CM]} Kolany's Patching Lemma. Assume that $\{A_j : j\in J\}$
is a family of non-empty sets and $\{ \Cal F_j : j\in J\}$ is a family of
non-empty sets of functions such that for every $j\in J$ and every
$f\in \Cal F_j$, dom $f = A_j$. Assume that for every finite
$J_0 \subseteq J$ there is a function $F_0$ such that for all $j\in J_0$,
$F_0\mid A_j \in \Cal F_j$, then there exists a function $F$ such that
for all $j\in J$, $F\mid A_j \in \Cal F_j$. \ac{Kolany} \cite{1999}.
\iput{Kolany patching lemma}
\item{}{\bf [43 W]} Countable products of compact Hausdorff spaces are
Baire. \ac{Brunner} \cite{1983c} and note 28.
\item{}{\bf [43 X]} Products of compact Hausdorff spaces are Baire.
\ac{Herrlich/Keremedis} \cite{1998} and note 28.
\item{}{\bf [43 Y]} Products of pseudo-compact spaces are Baire.
\ac{Herrlich/Keremedis} \cite{1998} and note 28.
\item{}{\bf [43 Z]} Products of countably compact, regular spaces are
Baire. \ac{Herrlich/Keremedis} \cite{1998} and note 28.
\item{}{\bf [43 AA]} Products of regular-closed spaces are Baire.
\ac{Herrlich/Keremedis} \cite{1998} and note 28.
\item{}{\bf [43 AB]} Products of \v Cech-complete spaces are Baire.
\ac{Herrlich/Keremedis} \cite{1998} and note 28.
\item{}{\bf [43 AC]} Products of pseudo-complete spaces are Baire.
\ac{Herrlich/Keremedis} \cite{1998} and note 28.
\bigskip
{\bf In note 28 replace the first definition with the following}
\medskip
\definition{Definition} Let $(X,T)$ be a topological space.
\item{1.} A set $Y\subseteq X$ is {\it nowhere dense} if the closure
of $Y$ has empty interior. \iput{nowhere dense set}
\item{2.} A set $Y\subseteq X$ is {\it meager} or {\it of the first
category} if $Y$ is a countable union of nowhere dense sets.
\iput{meager set} \iput{first category set}
\item{3.} A set $Y\subseteq X$ is {\it perfect} if $Y$ is closed,
non-empty and has no isolated points. \iput{perfect set}
\item{4.} A set $Y\subseteq X$ has the {\it Baire property} if
$(Y\setminus U)\cup(U\setminus Y)$ is meager for some open set $U$.
\iput{Baire property}
\item{5.} $(X,T)$ is {\it regular} if points are closed and every
neighborhood of a point contains a closed neighborhood of that point.
\iput{regular space}
\item{6.} A {\it 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
$F$. \iput{regular filter}
\item{7.} $(X,T)$ is {\it regular-closed} if $(X,T)$ is regular and
any regular filter on $X$ has a non-empty intersection.
\iput{regular-closed space}
\item{8.} $(X,T)$ is {\it Baire} or is a {\it Baire space}
if the intersection of each countable sequence of dense, open sets
in $X$ is dense in $X$. \iput{Baire space}
\item{9.} $(X,T)$ is {\it sequentially complete} if every sequence
has a convergent subsequence.
\item{10.} $(X,T)$ is {\it \v Cech complete-I} if it
satisfies \iput{\v Cech complete space}
\itemitem{(i)} $(X,T)$ is regular (every neighborhood of a point $x\in S$
contains a closed neighborhood of $x$) and
\itemitem{(ii)} 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.\par
\item{} (This is the definition of \v Cech complete used in form [43 E].)
\item{11.} $(X,T)$ is {\it \v Cech 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.)
\item{} (This is the definition of \v Cech complete used in $K9$ below.
$K9$ is equivalent to form 106.)
\item{12.} A collection $\Cal B$ of non-empty open sets is called a
{\it regular pseudo-base} for $(X,T)$ if:
\itemitem{(i)} For each non-empty $A\in T$, there is some $B\in\Cal B$
such that $cl B\subseteq A$ and
\itemitem{(ii)} If $A\ne\emptyset$ is in $T$ and $A\subseteq B$ for
some $B\in\Cal B$, then $A\in\Cal B$.
\item{13.} $(X,T)$ is {\it 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.
\iput{pseudo-complete space}
\item{14.} $(X,T)$ is {\it co-compact} if $\exists$ a family $F$ of
closed sets such that \iput{co-compact space}
\itemitem{(i)} If $G \subseteq F$ and $G$ has the finite intersection
property, then $\bigcap G \neq\emptyset$ and
\itemitem{(ii)} 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$({\Cal O}).)$
\item{15.} $(X,T)$ is {\it pseudo-compact} if every continuous real valued
function on $(X,T)$ is bounded. \iput{pseudo-compact space}
\item{16.} $(X,T)$ is {\it scattered} or {\it Boolean} if it is Hausdorff
and has a basis of clopen sets. \iput{scattered space} \iput{Boolean space}
\enddefinition
\end