数理解析研究所講究録
1531 巻 2007 年 49-54

49\v{C}ECH-COMPLETENESS IN FIBREWISE TOPOLOGY
島根大学総合理工学研究科 (首都師範大学) 白 云峰 (Yun-Feng Bai)
Department of Mathematics, Shimane Univ. (Capital Normal Univ.)
島根大学総合理工学部 三輪拓夫 (Takuo Miwa)
Department of Mathematics, Shimane University

1. INTRODUCTION-complete map”, and investigate some basic
We introduce a new notion
properties, invariance under perfect maps, relationships between (locally) compactmap, \v{C}ech-complete map and -map, and characterizations by compactifications of\v{C}ech-complete maps.
Motivation and significance of \v{C}ech-complete maps are:(1): The notion of \v{C}ech-complete maps in the fibrewise category $TOP_{B}$ is corresponding to the notion of \v{C}ech-complete spaces in the topological category TOP.
In fact, we can prove the following:-map.
-completecompact
Compact
Topology. So,\v{C}ech-completeinGeneralmaps is a new idea(2): The notion ofGeneral Topology becomes plentifully by this notion.
$‘(\check{\mathrm{C}}\mathrm{e}\mathrm{c}\mathrm{h}$$k$$\mathrm{m}\mathrm{a}\mathrm{p}\Rightarrow\check{\mathrm{C}}\mathrm{e}\mathrm{c}\mathrm{h}$$\mathrm{m}\mathrm{a}\mathrm{p}\Rightarrow \mathrm{L}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}$$\mathrm{m}\mathrm{a}\mathrm{p}\Rightarrow k$

For an arbitrary topological space one considers the category $TOP_{B}$ , the objectsof which are contin,uous maps into the space , and for the objects $p:Xarrow B$ andwith the, a morphism from into is a continuous map A ::is said:$parrow
q$property
A. This is denoted by A : $parrow q$ . A morphism. A
to be onto, closed, perfect, etc., if respectively, such is the map A :$X$continuous map $p:Xarrow B$ is called by a projection, and is called by a fibrewise
space over or (X, ) is called by a fibrewise space. Further we call :a morphism when we write ) : $parrow q$ , and we also call it a fibrewise map when we.
write
Throughout this paper, we assume that all spaces are topological spaces, and allmaps and projections are continuous. For other terminology and notations undefinedin this paper, one can consult [3] about TOP, and [5] and [8] about $TOP_{B}$ .
$B$

$B$

$q$

$\mathrm{Y}arrow B$

$p$

$q$

$Xarrow \mathrm{Y}$$\mathrm{A}$$p=q\mathrm{o}$$Xarrow \mathrm{Y}$$B$$\mathrm{A}$$p$$\mathrm{A}:(X,p)arrow(\mathrm{Y}, q)$$Xarrow \mathrm{Y}$50

2. PRELIMINARIESIn this section, we refer to the notions and notations in Fibrewise Topology.
Let $(B, \tau)$ be a fixed topological space with a fixed topology . Throughout
this paper, we will use the abbreviation $nbd(s)$ for neighborhood $(s)$ . We also use thatand are the sets of all
for $b\in B,$ $N(b)$ is the set of all open nbds of , andrespectively. Note that
numbers,natural numbers, all rational numbers and all real
regularity of $(B, \tau)$ is assumed in Theorems 3.7 and 3.10, further first countabilityof $(B, \tau)$ is as sumed in Theorem 3.10.
For a projection $p:Xarrow B$ and each point $b\in B$ , the fibre over is the subset$X_{b}=p^{-1}(b)$ of $X$ . Also for each subset $B’$ of
we regard $X_{B’}=p^{-1}(B’)$ as afibrewise space over $B’$ with the projection determined by . For a filter
is the filter generated by the family $\{p(F)|F\in F\}$ .
in $X$ , we denote thatFirst, we begin with defining some separation axioms on maps.
$B$$\tau$$b$$\mathrm{N},$$\mathrm{R}$$\mathrm{Q}$

$b$

$B$

$(\mathrm{b}\mathrm{a}s\mathrm{e})F$$p$$p_{*}(\mathcal{F})$: $Xarrow B$ is called a -map, $i=0,1,2$ , if for all$x\neqx’,$$p(x)=p(x’)$ the following condition is respectively$x’\in X$ such thatsatisfied:has a nbd in $X$ not containing the other(1) $i=0$ : at least one of the points
point;(2) $i=1$ : each of the points $x,x’$ has a nbd in $X$ not containing the other point;(3) $i=2$ : the points and have disjoint nbds in $X$ .

Deflnition 2.1. A projection$T_{1}$$p$$x,$$x,$

$x’$

$x’$

$x$

Definition 2.2. The subsets and $H$ of the space $X$ are said to be respectively:,(1) $nbd$ separated in,(2) functionally separated in
if the sets $F\cap U$ and $H\cap U$$F$$U\subset X$$U\subset X$(1) have disjoint nbds in ,(2) are functionally separated in (i.e. there exists a mapthat $F\cap U\subset\phi^{-1}(0)$ and $H\cap U\subset\phi^{-1}(1))$ .
$U$

$U$

$\phi$: $Uarrow[0,1]$ such

Deflnition 2.3. A projection $p:Xarrow B$ is called completely regular (resp. regular),, there exists
if for every point $x\in X$ and every closed set in $X$ such that
separatedfunctionallyareanda nbd $W\in N(p(x))$ , such that the sets
. A completely regular (resp. regular) -map is called(resp. nbd separated) in$\infty chonoff$ or) map.
(resp.
$F$$x\not\in F$$F$$\{x\}$$T_{0}$$X_{W}$$T_{3^{-}}$$T_{3_{3}^{1-}}$

It can be easily verified that every$i\leq j$.

$T_{j}$-map is a

$T_{i}$-map forDeflnition 2.4. Let $p:Xarrow B$ be a projection.
for all(1) The map is called a functionally$p(x)=p(x’)$ there exists $W\in N(b)$ such that the sets.
separated in$p$$T_{2^{-}}map.\mathrm{i}\mathrm{f}$$x,$ $x’\in$\{x\}$$X_{W}$$j,$ $i=0,1,2,3,3 \frac{1}{2}$

and

X$

and

such that $x\neq x’$ ,are functionally$\{x’\}$

51(2) The map is called functionally normal (resp. normal) if for every $b\in B$ andevery two disjoint, closed sets $F$ and $H$ in $X$ , there exists $W\in N(b)$ such that
$F$. A functionallyand $H$ are functionally separated (resp. nbd separated) in-map).
is called a functionally -map (resp.
normal (resp. normal)$p$$X_{W}$$T_{4}$$T_{4}$$T_{3^{rightarrow}}\mathrm{m}\mathrm{a}\mathrm{p}$It can be easily verified that (1) every -map is a -map, (2) every functionally-map is a-map and every-map is a functionally -map. However, note-map. For this, see the remark in thisthat every -map is not necessarily asection.
We now give the definitions of submaps, compact maps [9] and locally compactmaps [7].
$T_{3}$$T_{4}$$T_{2}$$T_{4}$$T_{3\frac{1}{2}}$$T_{3\frac{\iota}{2}}$$T_{4}$$T_{3\frac{1}{2}}$Definition 2.5. The restriction of the projection : $Xarrow B$ on a closed (resp.
open, type , etc.) subset of the space $X$ is called a closed (resp. open, type ,etc.) submap of the map .
$p$$G_{\delta}$$G_{\delta}$$p$Definition 2.6. (1) A projection $p:Xarrow B$ is called a compact map if it is perfect(i.e. it is closed and all its fibres $p^{-1}(b)$ are compact).
(2) A projection $p:Xarrow B$ is said to be a locally compa map if for each $x\in X_{b}$ ,where $b\in B$ , there exists a nbd $W\in N(b)$ and a nbd $U\subset X_{W}$ of such thatis:
is a compact map, where is the restriction of and.
the closure of in$ct$

$x$

$p’$

$X_{W}\cap\overline{U}arrow W$

$U$

$p’$

$X_{W}\cap\overline{U}$$p$$X_{W}$Note that a closed submap of a (resp. locally) compact map is (resp. locally)
compact, and for a (resp. locally) compact map $p:Xarrow B$ and every $B’\subset B$ therestriction $p|X_{B’}$ : $X_{B’}arrow B’$ is (resp. locally) compact.
is called aDefinition 2.7. (1) For a map : $Xarrow B$ , a map $c(p)$ :$X$
and $c(p)|X=p$ .
compactification of if $c(p)$ is compact, is dense in-compactifiable(2) A map $p:Xarrow B$ is called a -compactifiable map (resp.
and $c(p)$ is a -map (resp.
map) if has a compactification $c(p)$ :map).
$\mathrm{q}Xarrow B$$p$$\mathrm{q}X$$p$$T_{2}$$T_{3_{2}^{1}}$$c_{\mathrm{p}}Xarrow B$$p$$T_{2}$$T_{3\frac{1}{2}}-$

Remark: (1) The compactification of a map was studied by Pasynkov [7]. In
James [5] Section 8, there are some basic study of compactifiable maps, but notethat in [5] he uses a terminology “fibrewise compactification”. For other study ofcompactifiable maps, see [1] and [6].
-compactifiable maps because,(2) Note that we must consider bothandunlike the case of spaces, there exist -compact maps which are not –maps ([4]4.2 or [2] Example 3.4).
$T_{2^{-}}$$T_{3\frac{1}{2}}$$T_{3}$$T_{2}$

Deflnition 2.8. For the collection of fibrewise spaces
t.h subTychonoffof thespacespaces
with respectproductofthe:atheisproduct $\prod=\prod$ {} called fan.
to the maps$\{(X_{\alpha},p_{\alpha})|\alpha\in\Lambda\},$$X= \{t=\{t_{\alpha}\}\in\prod\{X_{\alpha} :$X_{\alpha}$$p_{\alpha},$$\alpha\in\Lambda$\alpha\in\Lambda :$\in\Lambda$

p_{\alpha}t_{\alpha}=p_{\beta}t_{\beta}\forall\alpha, \beta\in\Lambda\}$$X_{\alpha}$$\mathrm{e}$

52

, the
onto the factorof the product:For the projection
on $X$ will be called the projection of the fan product onto the factor
restriction
for,
A. IFlrom the definition of fan product we have that,every and in A. Thus one can define a map $p:Xarrow B$ , called the product of, and (X, ) is called the fibrewise $p7oduct$, bythe maps.
space of, are continuous.
Obviously, the projections and$pr_{\alpha}$$\prodarrow X_{\alpha}$$\prod$$X_{\alpha}$$\pi_{\alpha}$$X_{\alpha}$$\alpha\in$$p_{\alpha}\mathrm{o}\pi_{\alpha}=p_{\beta}\circ\pi_{\beta}$$\beta$$\alpha$$p_{\alpha},$$\alpha\in\Lambda$$p=p_{\alpha}\circ\pi_{\alpha},$$\alpha\in\Lambda$$p$$\{(X_{\alpha},p_{\alpha})|\alpha\in\Lambda\}$$\alpha\in\Lambda$$P$$\pi_{\alpha},$be a collection of fibrewise spaces.
Proposition 2.9. Let
is $T_{i}(i=0,1,2)$ (resp, functionally ), then the product is ailso(1) If each$(i=0,1,2)$ (resp. functionally ).
, then the product is also a)(resp.
(2) If eachis a surjec tive)(resp.
is a compact map, then the product is a compact map.
(3) If eachis a -compactifiable map, then the product is a -compactifiable(4) If each$\{(X_{\alpha},p_{\alpha})|\alpha\in\Lambda\}$$T_{2}$$p_{a}$$T_{i}$$p$$T_{2}$$T_{3^{-}}$$p_{\alpha}$$T_{3\#}-$$T_{3_{2}^{1}}-$$T_{3^{-}}$$p$$\mathrm{m}\mathrm{a}\mathrm{p}$$\mathrm{m}\mathrm{a}\mathrm{p}$$P$$p_{\alpha}$$p$$T_{2}$$p_{\alpha}$$T_{2}$map.

We shall conclude this section by defining the concept of b–filters (or tied filters)
which plays an important role in this report.
Definition 2.10. ([5] Section 4.) For a fibrewise space $X$ over , by a -filter (ortied filter) on $X$ we mean a pair $(b, F)$ , where $b\in B$ and $F$ is a filter on $X$ suchthat is a limit point of the filter $p_{*}(F)$ on . By an adherence point of a -filter $F$$(b\in B)$ on $X$ , we mean a point of the fibre
which is an adherence point of $F$ asa filter on $X$ .
$B$

$b$

$B$

$b$

$X_{b}$

3. SOME PROPERTIESDefinition 3.1. Let $X$ be a topological space, and a subset of $X$ . We say thatof subsets ofthe diameter of of the space $X$ is less than a family$s\in S$ such
$X$$\delta(A)<A$an,thereexists
provided
thatthe space , and we shall write.
thatis \v{C}ech-complete if for every
Definition 3.2. A -compactifiable map$b\in B$ , there exists a countable family
withof open (in $X$ ) covers of$F$forthe property that every -filter which contains sets of diameter less thanevery
has an adherence point.
isSince the real line with the usual topology is \v{C}ech-complete, :\v{C}ech-complete where is a one-point space. All rational numbers , as a subset
is open andis not \v{C}ech-complete though-complete, thusof , is not
closed. But we have the following results.
Theorem 3.3. For a \v{C}ech-complete map : $Xarrow B$ , if is closed subset of $X$ ,then $p|F:Farrow B$ is \v{C}ech-complete.
$A$

$A$

$A=\{A_{\mathit{8}}\}_{s\in S}$$A\subset A_{\partial}$$\mathrm{p}:Xarrow B$$T_{2}$$X_{b}$$\{A_{n}\}_{n\in N}$$A_{n}$$\mathrm{b}$$n\in \mathrm{N}$$p$$\mathrm{R}$$B$$\mathrm{R}$$\check{\mathrm{C}}\mathrm{e}\mathrm{c}\mathrm{h}$$\mathrm{Q}$$p|\mathrm{Q}$$p|\mathrm{Q}$

$p$

$F$

$\mathrm{R}arrow B$

53

Theorem 3.4. For a \v{C}ech-complete map $p:Xarrow B$ , if
$X$ is regular, then $p|G:Garrow B$ is \v{C}ech-complete.

$G$

is a

$G_{\delta}$-subset of $X$ andTheorem 3.5. Let $\{(X_{n},p_{n})|n\in N\}$ be a countable family of fibrewise spaces andbe the fibrewise product space. If every is surjective \v{C}ech-complete,then is \v{C}ech-complete.
be a perfect map, andTheorem 3.6. Let a fibrewise map A : (X, )and be -compactifiable maps. Then is \v{C}ech-complete if and only if is$( \prod_{B}X_{n},p)$$p_{n}$$\mathrm{P}$

$p$

$p$

$T_{2}$$q$$arrow(\mathrm{Y}, q)$

$p$

$q$

\v{C}ech-complete.

Theorem 3.7. Suppose that is regular. For a -compactifiable map $p:Xarrow B$ ,the following are equivalent:(1) $p:Xarrow B$ is \v{C}ech-complete.
is a(2) For every -compactipication: $X’arrow B$ of and each $b\in B,$
subset of .
is a -subset(3) There exists a -compactipication : $X’arrow B$ of such that$b\inB$.
offor each$B$

$T_{2}$$p’$$T_{2}$$X_{b}$$p$$G_{\delta^{-}}$$X_{b}’$$p’$$T_{2}$$X_{b}$$p$$G_{\delta}$$X_{b}’$

Theorem 3.8. Every locally compact map,

$T_{2}$-map is \v{C}ech-complete.

Definition 3.9. (James [5], Definitions 10.1 and 10.3) (1) Let (X, ) be a fibrewise
spsce. The subset $H$ of $X$ is quasi-open (resp. quasi-closed) if the following conditionis satisfied: for each $b\in B$ and $V\in N(b)$ there exists a nbd $W\in N(b)$ with $W\subset V$such that whenever $p|K:Karrow W$ is compact then $H\cap K$ is open (resp. closed) in$p$$K$.

map. The map is a -map if every: $Xarrow B$ be aquasi-closed subset of $X$ is closed in $X$ or, equivalently, if every quasi-open subsetof $X$ is open in $X$ .
is first countable and regular. Then a \v{C}echTheorem 3.10. Suppose thatcomplete map $p:Xarrow B$ is a k-map.
(2) Let a projection$T_{2}$$p$

$p$

$k$

$B$

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[3] R.Engelking: General Topology, Heldermann, Berlin, rev. ed., 1989.
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