数理解析研究所講究録
1531 巻 2007 年 113-128

113

Properties on relative paracompactness
and their absolute embeddings
奈良女子大学附属中等教育学校

慎二 (Shinji Kawaguchi)

川口Nara Women’s University Secondary School1.

Introduction

This report is a summary of [13].
and the symbol 7
Throughout this note all spaces are assumed to be
denotes an infinite cardinal. The symbol denotes the set of all natural numbers.
denote the closure and the interior
For a subset of a space $X,$andof in $X$ , respectively.
a subspace of $X$ . is Hausdorff (respectively,
Let $X$ be a space and(respectively,
and everystrongly Hausdorff) in $X$ if for every$X$$x\neqy$$x\in X)$ with, there exist disjoint open subsets
of such that $x\in U$
and $y\in$ .
is said to be regular (respectively, strongly regular) in $X$ if for,(respectively, $y\in X$ ) and each closed subset $F$ of $X$ with
each.
of $X$ such that $y\in U$ andthere exist disjoint open subsets
and each closed subset $F$
Moreover,is superregular in $X$ if for everyof $X$ such that $y\in U$, there exist disjoint open subsets
of $X$ withand $F\subset V$ ([1], [2] and [3]).
As relative notions of paracompactness, the following are known. Let $X$ bea space and a subspace of $X$ . For $x\in X$ , a collection $A$ of subsets of $X$ issaid to be locally finite at in $X$ if there exists a neighborhood of in $X$ whichintersects at most finitely many members of $A$ . In [1], [2] and [3], is said tobe 1- (respectively, 2-) paracompact in $X$ if for every open cover of $X$ , there)(respectively,
exists a collection of open subsets of $X$ withsuch that is a partial refinement of and is locally finite at each point of, there exists
in $X$ . Here, is said to be a partial refinement of if eachcontaining . We also say that is a refinement (respectively, an openarefinement, a closed refinement) of if is a cover (respectively, an open cover,a closed cover) of $X$ and a partial refinement of . The term “2-paracompact)’
is often simply said “paracompact”. Moreover, is said to be Aull-paracompact, there exists
in $X$ if for ev,e ry collection of open subsets of $X$ withis a partial
such thata collection of open subsets of $X$ withrefinement of and is locally finite at each point of in X. ([2], [4]). The
1-paracompactness and Aull-paracompactness of in $X$ need not imply each
$T_{\mathrm{i}}$$\mathrm{N}$$\overline{A}^{X}$$A$$\mathrm{I}\mathrm{n}\mathrm{t}_{X}A$$A$$\mathrm{Y}$$\mathrm{Y}$$x\in \mathrm{Y}$$y\in \mathrm{Y}$$U,$ $V$$V$$\mathrm{Y}$$y\not\in F$$y\in \mathrm{Y}$$F\cap \mathrm{Y}\subset V$$U,$ $V$$y\in \mathrm{Y}$$\mathrm{Y}$$U,$ $V$$y\not\in F$$\mathrm{Y}$

$x$

$x$

$\mathrm{Y}$$\mathcal{U}$$X=\cup \mathcal{V}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathrm{Y}$$\mathcal{U}$$\mathcal{V}$$\mathcal{V}$$V\in \mathcal{V}$$\mathcal{U}$$\mathcal{V}$$V$$U\in \mathcal{U}$$\mathcal{V}$$\mathcal{U}$$\mathcal{V}$$\mathcal{U}$$\mathrm{Y}$$\mathrm{Y}\subset\cup \mathcal{U}$$\mathcal{U}$$\mathcal{U}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathcal{V}$$\mathrm{Y}$$\mathcal{V}$$\mathrm{Y}$114
other ([4]), but each of them clearly implies 2-paracompactness of in $X$ . Whenis a closed subspace of $X,$ is 2-paracompact in $X$ if and only if is Aullparacompact in $X$ .
Aull [5] defined that is -paracompact in $X$ if for every collection of, there exists a collection of open subsets ofopen subsets of $X$ with
$X$ such thatis a partial refinement of and is locally finite in
X. Recall that 1- and a-paracompactness do not imply each other in general.
But for a regular space $X$ , if is a-paracompact in $X$ then is l-paracompact
in $X$ , the converse also holds if, in addition, is closed ([17, Theorem 1.3], seealso Proposition 3.1 below for a generalization).
These notions are central in the study of relative paracompactness and thefollowing rela.t ions fold.
$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathcal{U}$$\mathrm{Y}$$\alpha$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{U}$$\mathrm{Y}\subset\cup \mathcal{V},$$\mathcal{U}$$\mathcal{V}$$\mathcal{V}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$is 1-paracompact in $X$$\downarrow$$\mathrm{Y}$is 2-paracompact in $X$$\dagger$$\mathrm{Y}$is Aull-paracompact in $X$$1$$\mathrm{Y}$is a-paracompact in $X$

DIAGRAM 1
Moreover, absolute embeddings of above relative paracompactness are characterized as follows (see also [14]).

-Outerelo [17]). For a( -spectively, regular) space , the following statements are equivalent.
is l-(or equivalently, ) paracompact in every larger Tychonoff (respectively, regular) space.
(respecis l-(or equivalently, ) paracompact in every larger
tively, regular) space containing as a closed subspace.
is compact.

Theorem 1.1 ($re$$\mathcal{T}ychonoff$$\mathrm{L}\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{a}\tilde{n}\mathrm{e}\mathrm{z}[15];\mathrm{L}\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{a}\overline{\mathrm{n}}\mathrm{e}\mathrm{z}$$\mathrm{Y}$$(a)\mathrm{Y}$$\alpha-$$(b)\mathrm{Y}$$\alpha-$$\mathcal{I}ychonoff$$\mathrm{Y}$$(c)\mathrm{Y}$Theorem 1.2 (Arhangel’skfi-Genedi [3]; see also [9], [20]). For a $\infty chonoff$(respectively, regular) space , the following statements are equivalent.
is 2- (or equivalently, Aull-) paracompact in every larger $\Phi chonoff$(respec-tivdy, regular) space.
is 2- (or equivalently, Aull-) paracompact in every larger Tychonoff
as a closed subspace.
(respec-tively, regular) space containing$\mathrm{Y}$$(a)\mathrm{Y}$$(b)\mathrm{Y}$$\mathrm{Y}$

115
(c)

$\mathrm{Y}$is.

$Lindel\dot{\mathit{0}}f$[ , page 98], [2, page 174] asked if one can generalize the notions
above to the well known Michael’s criteria of paracompactness in [18] and [19].
Concerning this problem, Aull [6, Theorem 5] already proved that a subspaceof a normal space $X$ is a-paracompact if and only if for every cover of byopen subsets of $X$ has a closure-preserving partial open refinement which covers[ $16$ , Theorem 1.3] proved that a subspace of a regular
Y. Moreover,space $X$ is a-paracompact if and only if for every cover of by open subsets
of $X$ has a partial refinement (or equivalently, a closed partial refinement) $A$ of.
such that $A$ is locally finite in $X$ and
In Section 2, we introduce notions of relative paracompactness by using locally finite (not necessarily open) partial refinement and locally finite closed
partial refinement. We also consider closure-preserving cases.
In Section 3, we discuss locally finite open refinement and closure-preserving, whereis a space obtained from $X$
open refinement by using the space
be isolated.
by letting each point of
In Section 4, we investigate their basic properties and discuss their absoluteembeddings. In particular, we have$\mathrm{A}\mathrm{r}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{l}’ \mathrm{s}\mathrm{k}\mathrm{i}\mathrm{l}$$1$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{L}\mathrm{u}\mathrm{p}\mathrm{i}\mathrm{a}\tilde{\mathrm{n}}\mathrm{e}\mathrm{z}$$\mathrm{Y}$$\mathcal{U}$$Y\subset \mathrm{I}\mathrm{n}\mathrm{t}_{X}(\cup A)$$\mathcal{U}$$X_{\mathrm{Y}}$$X_{\mathrm{Y}}$$X\backslash \mathrm{Y}$(respectively, regular) space , the following
Theorem 1.3. For astatements are equivalent.
l-lf- (or equivalently, l-cp-) paracompact in $eve\eta$ larger Tychonoff(respectively, regular) space.
is l-lf- (or equivalently, l-cp-) paracompact in every larger $7ychonoff$subspace.
(respectively, regular) space containing as a$(c)Y$ is Lindel\"of.
$\mathcal{I}ychonoff$$\mathrm{Y}$$(a)\mathrm{Y}\dot{u}$$(b)\mathrm{Y}$$\mathrm{Y}$$clo\mathit{8}ed$

Theorem 1.4. A $\tau ychonoff$ (respectively, regular) space is a-f- (or equiva) paracompact in every larger Tychonoff (respectively, regular) space
lently,if and only if is compact.
$\mathrm{Y}$$\alpha- cp-$$\mathrm{Y}$in Theorem 1.1 cannot becase, a similar statement toForadded to Theorem 1.4. Indeed, we replace “every larger Tychonoff (respectively,
larger Tychonoff (respectively, regular) space containregular) space” by“
ing as a closed subspace” in Theorem 1.4, “ is compact” is replaced byis paracompact” (see Remark 4.4). In addition, we point out that a Tychonoff
is 2- (or equivalently, Aull-) .paracompact in(respectively, regular) space
every larger Tychonoff (respectively, regular) space if and only if is paracompact (see Theorem 4.5 and Remark 4.6).
In the final section, a remark on definitions of relative paracompactness due
to Grabner et.al. [10], [12] will be given and a gap of a result in [11] will bepointed out.
$(b)$$\alpha- cp\cdot \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}$$‘(\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{c}\mathrm{p}$$\mathrm{Y}$116
For general surveys on relative topological properties, see thesubsequent articles [1] and [2]. Other undefined notations and terminology areused as in [7] and [13].
$\mathrm{A}\mathrm{r}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{l}’ \mathrm{s}\mathrm{k}\mathrm{i}\mathrm{l}’ \mathrm{S}$Some versions of relative paracompactness

2

In this section, we newly define some notions of relative paracompactness
and discuss their basic properties.
Let $X$ be a space and a subspace of $X$ . We define that is l-lf-paracompact(respectively, l-lfc-paracompact) in $X$ if every open cover of $X$ has a refinement(respectively, a closed refinement) of which is locally finite at each point of
in $X$ . We also define that is 2-lf-paracomPact (respectively, 2- fc-paracompact)
in $X$ if for every open cover of $X$ there exists a partial refinement (respectively,
and is locally finite at eacha closed partial refinement) such that
is Aull-lf-paracompact (respectively, Aull-lfcpoint of in $X$ . Furthermore,,paracompact) in $X$ if for every collection of open subsets of $X$ withthere exists a partial refinement (respectively, a closed partial refinement) of
and is locally finite at each point of Y. We also say
such that-paracompact) in $X$ if for every-paracompact (respectively,that isthere exists a partial refinement
collection of open subsets of $X$ withand is(respectively, a closed partial refinement) of such that$X$
locally finite in .
Let $X$ be a space and $x\in X$ . A collection $A$ of subsets of $X$ is said to be$closure- presewi_{\backslash }ng$ at, it holds that
in $X$ if for every $A’\subset A$ with. The following are known.
, where$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathcal{U}$$\mathrm{Y}$$\mathcal{U}$$Y$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathcal{V}$$\mathrm{Y}$$\mathrm{Y}\subset\cup \mathcal{U}$$\mathcal{U}$$\mathcal{V}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathcal{U}$$Y$$\alpha- lfc$$\alpha- lf$$\mathrm{Y}\subset\cup \mathcal{U}$$\mathcal{U}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathcal{U}$$\mathcal{V}$$x\in\overline{\cup A^{\prime^{X}}}$$x$$x\in\cup\overline{A^{\prime^{X}}}$$\overline{A^{\prime^{X}}}=\{\overline{A}^{X}|A\in A’\}$

Proposition 2.1. For a collection $A$ of subsets of a space $X$ and $x\in X$ , eachof the following statements hold.
$(a)$
If $A$ is locally finite at in $X$ , then $A$ is closure-preserwing at in $X$ .
$A$ is locallyfinite (respectively, closure-preserving) at in $X$ if and onlyis also locally finite (respectively, closure-preseruing) at in $X$ .
$(c)$ $A$ is locallyis point-finite at and $A$ isfinite at in $X$ if and only
closure-preserving at in $X$ .
$x$

$x$

$x$

$(b)$

$if\overline{A}^{X}$$x$$if\overline{A}^{X}$

$x$

$x$

$x$

Hence, we have the following: $(a’)$ If $A$ is locally finite at each point of in$X$ , then $A$ is closure-preserving at
ach point of in X. $(b’)$ If $A$ is closureis also closure-preserving at eachpreserving at each point of in $X,$ thenpoint of in X. $(d)$ For a collection $A$ of closed subsets of $X,$ $A$ is locally finite
at each point of in $X$ if and only if $A$ is point-finite at each point of $Y$ andclosure-preserving at each point of in $X$ . Grabner et.al. [10], [12] introduced$\mathrm{Y}$$\mathrm{Y}$$e$$\overline{A}^{X}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$117
some relative notions related to closure-preserving collections; but their notions
do not necessarily satisfy any of $(a’),$and $(c’)$ above (for detail, see Section$(b’)$

5).

Let $X$ be a space and a subspace of $X$ . We define that is l-cp-paracompact(respectively, l-cpo-paracompact, l-cpc-paracompact) in $X$ if every open coverof $X$ has a refinement (respectively, an open refinement, a closed refinement)
which is closure-preserving at each point of
in $X$ . We also define thatis 2-cp-paracompact (respectively, 2-cpo-paracompact, 2-cpc-paracompact) in $X$
if for every open cover of $X$ there exists a partial refinement (respectively,
an open partial refinement, a closed partial refinement) such thatand is closure-preserving at each point of in $X$ (see Remark 5.1 below). We
say that is Aull-cp-paracompact (respectively, Aull-cpo-paracompact, Aull-cpcparacompact) in $X$ if for every collection of open subsets of $X$ withthere exists a partial refinement (respectively, an open partial refinement, aclosed partial refinement) such thatand is closure-preserving at$X$-paracompact (respeceach point of in . Moreover, we say that
is
$X$
of
-paracompact) in if for every-paracompact,
tively,$X$
open subsets of withthere exists a partial refinement (respectively,
an open partial refinement, a closed partial refinement) such that
$Y$

$Y$

$Y$

$\mathrm{Y}$$\mathcal{U}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$Y$$\mathcal{V}$$\mathrm{Y}$$\mathrm{Y}\subset\cup \mathcal{U}$$\mathcal{U}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathrm{Y}$$\mathcal{V}$$\mathrm{Y}$$\alpha- cpo$$\alpha- cp$$\mathrm{c}\mathrm{o}\mathrm{U}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathcal{U}$$\alpha- cpc$$\mathrm{Y}\subset\cup \mathcal{U}$$\mathcal{V}$$\mathrm{Y}\subset\cup \mathcal{V}$and is closure-preserving in $X$ .
Proposition 2.1 $(b)$ induces the following.
$V$

Proposition 2.2. Let be a subspace of a regular space X. Then, each of thefollouring statements hold.
$(a)$
If is l-lf-paracompact in $X$ , then is l-lfc-paracompact in $X$ .
$(b)$
If is l-cp-paracompact in $X$ , then is l-cpc-paracompact in $X$ .
$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$Remark 2.3. If we replace ‘(1-“ by “ “ “2-“ or “Aull-,, in the statements $(a)$andof Proposition 2.2, then the condition “X is regular” can be weakened“
tois strongly regular in $X$ ”.
$\alpha-$$(b)$$\mathrm{Y}$For closed subspaces, we have the following. Here, notice that 2-cpc-pax acompactness of in $X$ induces regularity of when is closed in $X$ .
$\mathrm{Y}$$\mathrm{Y}$Theorem 2.4. For a closed subspaceare equivalent.
-paracompact in $X$ .
isis 2-cpc-paracompact in $X$ .
-paracompact in $X$ andis$(d)Y$ is 2-cp-paracompa in $X$ andis paracompact Hausdorff.
$(a)\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$of a space$\alpha- lfc$$(b)\mathrm{Y}$$(c)\mathrm{Y}$$\alpha- lf$$ct$$(e)\mathrm{Y}$is regular.
$Y$
is regular.

$\mathrm{Y}$$X$, the following statements

118
Aull [5] proved that if a subspace of a Hausdorff space $X$ is -paracompact
in $X$ then is closed in $X$ . We improve this fact as follows.
$\mathrm{Y}$$\alpha$$Y$

Lemma 2.5. Assume that is strongly Hausdorff in X. Ifpact in $X$ , then is closed in $X$ .
$\mathrm{Y}$$\mathrm{Y}$is$\alpha- cp$-paracom-$\mathrm{Y}$The following corollary immediately follows from Theorem 2.4 and Lemma 2.5.
Corollary 2.6. Assume that is strongly Hausdorff in X. Then, each of thefollowing statements hold.
-paracompact in $X$ .
-paracompact in $X$ if and only if isis$(b)$ Assume that-pamcompact in $X$ if and onlyisis regular. Then,-paracompact in $X$ .
if is$\mathrm{Y}$$\mathrm{Y}$$\alpha- lfc$$(a)\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\alpha- cpc$$\alpha- lf$$\alpha- cp$(respectively, ) denotes the class of all regular
Hereafter, the symbol
and(respectively, Hausdorff) spaces. Moreover, the symbols““
is regular in $X$ ”, “ isis strongly Hausdorff in $X$ ”,
mean the conditions
superregular in $X$ ” and “$Y$ is strongly regular in $X$ ”, respectively. The symbol$\mathcal{T}_{2}$$\mathcal{T}_{3}$$\mathrm{S}\mathrm{H},$$\mathrm{R},$$\mathrm{S}\mathrm{t}\mathrm{R}$$\mathrm{S}\mathrm{u}\mathrm{R}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$“isdenotes the family of all closed subsets of $X$ . We denote the condition”-embedded in $X$ (see Section 3 for definition) by T3.
The following implications around 1-paracompactness follow from definitions, Proposition 2.2 and Theorem 3.4 below.
$\mathrm{Y}$$C_{X}$$T_{3}$$\mathrm{Y}$is$\mathrm{Y}$is$\mathrm{X}\in \mathcal{T}_{3}$is-paracompact$\mathrm{Y}$$1- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}-1- lf- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=1- lfc$$\mathrm{T}_{3}\mathrm{R}|\downarrow \mathrm{i}\mathrm{n}X$$\mathrm{Y}$is$\mathrm{i}\mathrm{n}_{1^{\mathrm{x}}}$$\mathrm{i}\mathrm{n}_{1^{x}}$

$Y$

is

X

$\in \mathcal{T}_{3}$$1- cpo- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}-1- cp- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=1- cpc$

in $X$

in $X$

is-paracompact
in $X$$\mathrm{Y}$DIAGRAM 2

For -paracompact case, we have the following implications. Thes implications directly follow from definitions, Corollary 2.5, Remark 2.3 and Theo$\alpha$

rem 3.5.

$e$

119
is

$\mathrm{Y}$

is

$Y$

$Y$

$\mathrm{S}\mathrm{t}\mathrm{R}$is-paracompact
in $X$$\alpha- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}-\alpha- lf- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=\alpha- lfc$

in $X$$\mathrm{S}\mathrm{u}\mathrm{R}$li

in $X$

SH$\mathrm{Y}\in \mathcal{T}_{3}|\mathrm{S}\mathrm{H}\downarrow$I$\alpha- cp\not\in \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}arrow\alpha- cp\cdot \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=\alpha- cpc- \mathrm{p}.\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}Y\mathrm{i}\mathrm{s}\mathrm{Y}\mathrm{i}\mathrm{s}\mathrm{S}\mathrm{t}\mathrm{R}\mathrm{Y}\mathrm{i}\mathrm{s}$in $X$

in $X$$\mathrm{m}X$

DIAGRAM 3
Moreover, the following implications hold for 2-paracompact case. These implications follows from definitions, Theorem 2.4, Remark 2.2 and Theorem 3.3
below.
isis
is2-paracompact–2-lf-paracompact $=2- lfc$-paracompact
in $X$
in $X$
in $X$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}\in C_{\mathrm{X}}|\mathrm{R}\downarrow$$\mathrm{Y}$$\mathrm{Y}\in C_{\mathrm{X}}1\downarrow$$\mathrm{Y}\in c_{\mathrm{x}}\mathrm{Y}\in \mathcal{T}_{3}|\downarrow$is$\mathrm{Y}$is$\mathrm{S}\mathrm{t}\mathrm{R}$$2- cp\triangleright \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}-2- cp\cdot \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=$in $X$$\mathrm{Y}$$\mathrm{S}\mathrm{t}\mathrm{R}$in $X$

is2-cpc-paracompact
in $X$$\mathrm{Y}$DIAGRAM 4
Finally, for Aull-paracompact case, we have the following implications. These
implications follow from definitions, Theorem 2.4, Remark 2.3 and Theorem 3.2.
$\mathrm{Y}$is$\mathrm{Y}$isin $X$$\mathrm{R}1\downarrow$

in $X$$\mathrm{Y}\in C_{\mathrm{X}}\mathrm{Y}\in \mathcal{T}_{3}|\downarrow$is-paracompact
in $X$$\mathrm{Y}$$\mathrm{A}\mathrm{u}\mathrm{l}\mathrm{l}- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}rightarrow \mathrm{A}\mathrm{u}\mathrm{U}- lf- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=\mathrm{A}\mathrm{u}\mathrm{l}\mathrm{l}- lfc\mathrm{S}\mathrm{t}\mathrm{R}$$\mathrm{Y}\in C_{\mathrm{X}}|\downarrow$$\mathrm{A}\mathrm{u}\mathrm{l}1- cpr\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}-\mathrm{A}\mathrm{u}\mathrm{l}1- c\mathrm{p}\cdot \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}_{-}^{arrow}\mathrm{A}\mathrm{u}\mathrm{l}1- cpc- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}Y\mathrm{i}\mathrm{s}\mathrm{Y}\mathrm{i}\mathrm{s}\mathrm{S}\mathrm{t}\mathrm{R}Y\mathrm{i}\mathrm{s}$in $X$

in $X$DIAGRAM 5

in $X$

120“, “2-“ and “Aull-,, can be replaced by
In Diagram 1, the terms “1-)’,““, “2-lf-“ and “Aull-lf-“, respectively. Moreover, these terms can be“l-lf-,,,“, ‘(2-lfc-“ and “Aull-lfc-“, respectively. Furthermore,
replaced by “l-lfc-,,, “
the same is available for cpo-, cp- and cpc-.
Let us emphasize the following proposition.
“$\alpha-$$\alpha- lf-$$\alpha- lfc-$

Proposition 2.7. Let be a subspace of a space X. If$X$ , thenis l-lf-paracompact in $X$ .
$\mathrm{Y}$$Y$

is 2-paracompact in$Y$For reverse implications in Diagrams 2, 3, 4, and 5, we have the following
examples.

Example 2.8. There exist a Tychonoff space $X$ and its closed subspace suchthat is a-lf-paracompact in $X$ , but not l-cp-paxacompact in $X$ (hence, not
2-paracompact in $X$ ).
$\mathrm{Y}$$\mathrm{Y}$

Example 2.9. There exist a Tychonoff space $X$ and its closed subspace such
is Aull-paracompact in $X$ , but not 1-paracompact in $X$ (hence, $Y$ isthatl-lf-paracomPact in $X$ , but not -paracompact in $X$ ).
$\mathrm{Y}$$\mathrm{Y}$$\alpha$3l-cpo-, $2-cpo-$ , Aull-c$po-$ and $\alpha-cpo$ -paracompactnessof a subspace in a space) embedded in $X$ if for every closed subset(respectively,
is said to be$F$), (respectively, ) and areof $X$ disjoint from (respectively,$X$separated by disjoint open subsets of ([5], see also [14]).
We often use the following proposition.
$\mathrm{Y}$$T_{3^{-}}$$T_{4^{-}}$$z\in X\backslash \mathrm{Y}$$\mathrm{Y}$

$F$

$z$

$\mathrm{Y}$Proposition 3.1 ([14]; see also [5], [17]). Let be a subspace of a space $X$ .
Then, the following statements are equivalent.
is 1-paracompact in $X$ and -embedded in $X$ .
is 2-paracompact in $X$ and -embedded in $X$ .
is Aull-paracompact in $X$ and -embedded in $X$ .
$(d)Y$ is a-paracompact in $X$ and satisfies the following condition $(*)$ : for, there exists
every $y\in Y$ and every closed subset $F$ of $X$ with.
an open subset of $X$ such that$\mathrm{Y}$$(a)\mathrm{Y}$$T_{3}$$(b)\mathrm{Y}$$T_{4}$$T_{4}$$(c)\mathrm{Y}$$F\cap \mathrm{Y}=\emptyset$$y\in U\subset\overline{U}^{X}\subset X\backslash F$$U$

As was stated in the previous section, we haveis regular in X. Then,
Theorem 3.2. Assume that
in $X$ if and only if is Aull-cpo-paracomPact in $X$ .
$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$is Aull-paracompact

121
is a closed subspace of $X$ andis regular in
Theorem 3.3. Assume that$X$
X. Then,is 2-paracompact in
if and only if is 2-cpo-paracompact in $X$ .
$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$is regular in $X$ and -embedded in X. Then,
Theorem 3.4. Assume thatis 1-paracompact in $X$ if and only if is l-cpo-paracompact in $X$ .
$Y$$T_{3}$$Y$$\mathrm{Y}$In Theorem 3.4, the condition “ is -embedded in $X$ “ cannot be removed.
Consider $X$ as the space $\Psi=\omega\cup A$ constructing a m.a.d. family $A$ of infinite.
andsubsets of$\mathrm{Y}$$T_{3}$$\mathrm{Y}=\omega$$\omega([8,5\mathrm{I}])$

Theorem 3.5. Assume that is superregular in $X$ (more generally, satisfies
is -paracompact in $X$ if andthe condition $(*)$ in Proposition $3.1(d))$ . Then,-paracompact in $X$ .
only if is$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\alpha$$\alpha- cpo$

Theorem 3.5 is a generalization of [6, Theorem 5] where $X$ is normal.
the space $X$ , with the topology
denote the space obtainedLet
generated by a subbase { $U|U$ is open in $X$ or}. Hence, points$X$
generateand. Moreover,inare isolated and is closed in
is often useful. As is seen in [1], the space
the same topology onin discussing several relative topological properties. It is easy to see that is
is Hausdorff (respectively,
Hausdorff (respectively, regular) in $X$ if and only if$\mathrm{h}\mathrm{o}\mathrm{m}$$X_{\mathrm{Y}}$$U\subset X\backslash \mathrm{Y}$$X_{Y}$$\mathrm{Y}$$X\backslash \mathrm{Y}$$X_{\mathrm{Y}}$$X_{Y}$$\mathrm{Y}([7])$$\mathrm{Y}$$X_{\mathrm{Y}}$regular).

Lemma 3.6. Let be a subspace of a space X. Then, is Aull-cpo-paracomhas a closure-preserving open
pact in $X$ if and only if every open cover of$Y$$\mathrm{Y}$$X_{\mathrm{Y}}$refinement.

To prove Theorems 3.4 and 3.5, we have the following lemma which improves[17, Lemma 1.2].

Lemma 3.7. For a subspace of a space $X$ , each of the following statements
hold.
$(a)$
If is -embedded in $X$ and l-cpo-paracompact in $X$ , then isembedded in $X$ .
Assume that satisfies the condition $(*)$ in Proposition $3.1(d)$ . If is-paracompact in $X$ , then is -embedded in $X$ .
$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$(b)$$T_{3}$$T_{4^{-}}$$Y$$\mathrm{Y}$$\mathrm{Y}$$\alpha- cpo$

$T_{4}$

Corresponding to Proposition 3.1, we have the following result for cpoparacompact cases. This fact follows from Theorems 3.2, 3.3, 3.4 and 3.5, Proposition 3.1
and Lemma 3.7. Notice that if is superregular in $X$ , then obviously satisfies
the condition $‘(*)$ in Proposition 3.1 $(d)$ .
$\mathrm{Y}$Corollary 3.8. Let
ments are equivalent.

$\mathrm{Y}$$\mathrm{Y}$be a subspace of a space X. Then, the following state-122$(a)\mathrm{Y}$$(b)\mathrm{Y}$$(c)\mathrm{Y}$is l-cpo-paracompact in $X$ and -embedded in $X$ .
is 2-cpo-paracompact in $X$ and -embedded in $X$ .
is Aull-cpo-paracompact in $X$ and -embedded in $X$ .
$T_{3}$$T_{4}$$T_{4}$At the end of this section, we discuss absolute embeddings of 1-,
and Aull-cpo-paracompactness. Corollary 3.9 below immediately follows from
Theorems 1.1, 3.4 and 3.5.
$\alpha-,$$2-$

Corolary 3.9. For a $\tau ychonoff$ (respectively, regular) space , the following
statements are equivalent.
) paracompact in $eve\eta$ larger
is l-cpo-(or equivalently,(respectively, regular) space.
) paracompact in $eve\eta$ larger $\tau ychonoff$is l-cpo-(or equivalently,(respectively, regular) space containing as a closed subspace.
$\mathrm{Y}$$(a)\mathrm{Y}$$\alpha- cpo-$$(b)\mathrm{Y}$$\alpha- cpo-$$\mathbb{R}chonoff$$Y$$(c)\mathrm{Y}$is compact.

Theorems 1.2, 3.2 and 3.3 induce the following.

Corollary 3.10. For a Tychonoff (respectively, regular) space , the following
statements are equivalent.
is 2-cpo- (or equivalently, Aull-cpo-) paracompact in $eve\eta$ larger
chonoff (respectively, regular) space.
is 2-cpo- (or equivalently, Aull-cpo-) paracompact in $eve\eta$ larger
chonoff (respectively, regular) space containing as a closed subspace.
$\mathrm{Y}$$(a)\mathrm{Y}$$\mathbb{R}-$$(b)\mathrm{Y}$$\mathcal{I}U-$$\mathrm{Y}$$(c)\mathrm{Y}$4

is $Lindel\dot{o}f$.

More on absolute embeddings

In this section, we discuss absolute embeddings on other versions of relative
paracompactness defined in Section 2. The results obtained in this section should
be compared with Theorems 1.1 and 1.2.
We actually give characterizations of absolute l-lf- and l-cp-paracompactness
as follows.
Theorem 4.1. For a Tychonoff (respectively, regular) space , the following
statements are equivalent.
is l-lfc-paracompact in every larger $\infty chonoff$ (respectively, regular)
space.
$(b)Y$ is l-cpc-paracompact in every larger Rchonoff (respectively, regular)
space.
$\mathrm{Y}$$(a)\mathrm{Y}$123$(c)\mathrm{Y}$is l-lf-paracompact in every larger Tychonoff (respectively, regular) space.

$(d)\mathrm{Y}$is l-cp-paracompact in every larger Tychonoff (respectively, regular)

space.

is.
In the statements from $(a)$ to $(d)$ above, “every larger Tychonoff (respectively,
regular) space“ can be replaced bylarger(respectively, regular),$Y$space containing as a closed subspace.
$(e)\mathrm{Y}$$Lindel\dot{\mathit{0}}f$$\mathcal{I}ychonoff$$” eve\eta_{J}$The proof of Theorem 4.1 is based on the following fact: let $X=A(\omega_{1})\cross$and. Then, is not l-cpparacompact in $X$ .
$\mathrm{Y}=(\{\infty\}\cross\omega)\cup(D(\omega_{1})\cross\{\omega\})$$(\omega+1)\backslash \{\langle\infty,\omega\rangle\}$$\mathrm{Y}$

Example 4.2. There exist a Tychonoff space $X$ and an open subspace of $X$
such that is Aull-paracompact in $X$ and 1-cpoparacompact in $X$ , but neither
l-paracompact in $X$ nor -crParacompact in $X$ .
$\mathrm{Y}$$\mathrm{Y}$$\alpha$For absolute a-lf- or-paracompactness, we have$\alpha- cp-$

Theorem 4.3. For astatements are equivalent.

$\mathbb{R}chonoff$$(a)\mathrm{Y}$(respectively, regular) space$\mathrm{Y}$, the following

is a-lfc-paracompact in every larger $\tau ychonoff$ (respectively, regular)

space.
-paracompact in $eve\eta$ larger(respectively, regular)isspace.
$(c)Y$ is $\alpha- lf- para\omega mpact$ in every larger $\tau ychonoff$ (respectively, regular) space.
$(b)\mathrm{Y}$$\mathbb{R}chonoff$$\alpha- cpc$

-paracompact in $eve\eta$ larger Rchonoff (respectively, regular)isspace.
is compact.

$(d)Y$$\alpha- cp$$(e)\mathrm{Y}$Remark 4.4. Notice that in Theorems 4.3, “every larger Tychonoff (respectively, regular) space” cannot be replaced by “every larger Tychonoff (respectively, regular) space containing as a closed subspace”. Indeed, for a Tychonoff(respectively, regular) space , the following statements are equivalent:-paracompact in every larger Tychonoff (respectively, regular)isspace containing as a closed subspace.
isin every larger Tychonoff (respectively, regular)
space containing as a closed subspace.
is paracompact.
“) can(or equivalently, “In the statements $(a)$ and $(b)$ above, “
“
“
“
(or
“).
be replaced by$\mathrm{Y}$$\mathrm{Y}$$(a)\mathrm{Y}$$\alpha- lfc$$\mathrm{Y}$$(b)\mathrm{Y}$$\alpha- cp\cdot \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}$$\mathrm{Y}$$(c)\mathrm{Y}$$\alpha- lfc-"$$\alpha- lf-$$\alpha- cpc-$$\alpha- cp-$

124

Moreover, we characterize absolute embeddings of relative paracompactnessof 2- or Aull-paracompactness types as follows.

Theorem 4.5. For a Tychonoff (respectively, regular) space , the following
statements are equivalent.
is Aull- $tfc$-paracompact in every larger Tychonoff (respectively, regular)
space.
is 2-cp-paracompact in every larger $\tau bchonoff$ (respectively, regular)
space.
is paracompact.
In the statements $(a)$ and $(b)$ above, “eve larger Tychonoff (respectively,(respectively, regular)
regular) space“ can be replaced by $” every,$, larger
space containing as a closed subspace”.
$\mathrm{Y}$$(a)\mathrm{Y}$$(b)\mathrm{Y}$$(c)\mathrm{Y}$$\mathrm{r}y$$\mathbb{R}chonoff$$\mathrm{Y}$Remark 4.6. Theorem 4.5 shows that “Aull-lfc-paracompact” in Theorem 4.5
can be replaced by “Aull-cpc-paracompact”, “Aull-lf-paracompact” and “Aullcp–paracompact”. Moreover in Theorem 4.5, “2-cp–paracompact” can be replaced by “2-lfc-paracompact”, “2-cpc-paracompact” and “2-lf-paracompact”.

5

Concluding remarks

In this section, we give some related remarks to relative paracompactness
discussed in the previous sections. Let be a subspace of a space $X$ and $F$ acollection of subsets of $X$ . In [10] ans [12], Grabner et.al. introduced the following
two relative notions of closure-preserving collections. It is defined in [12] that $F$ iseitherclosure preserving with respect to if for every$X$,7‘
is closed in . Moreover, is weakly closure preserving withrespect to if for every $F’\subset\{F\in F|F\cap Y\neq\emptyset\}$ , it holds that
. In [10], they assume that .7‘ is a collection of closed subsets of $X$
in the above definitions. As was mentioned in Section 2, the notion of closure
preserving collections with respect to above does not satisfy the statements$(a’),$ $(b’)$ and $(d)$ stated below Proposition 2.1. Actually, there exists a collection$A$ of closed subsets of $X$ such that $A$ is locally finite at each point ofin$X$ , but not closure preserving with respect to(consider $X=\omega+1,$
and $A=\{\{n\}|n<\omega\})$ . There exists a collection $A$ of subsets of $X$ such that
$A$ is closure preserving with respect to, but
is not closure preserving
andwith respect to (consider,$A=\{\{n\}\cross\omega|n<\omega\})$ . Moreover, there exists a collection $A$ of closed subsets of$X$ which is point-finite at each point ofand closure preserving with respect to, but not locally finite at some point of in $X$ (consider $X=\omega+1,$
and $A=\{\{n\}|n<\omega\})$ .
$Y$$F’\subset\{F\in F|F\cap \mathrm{Y}\neq\emptyset\}$$\mathrm{Y}$$\mathrm{Y}\subset\cup F’\mathrm{o}\mathrm{r}\cup \mathcal{F}’$$(\cup F’)\cap \mathrm{Y}=$$\mathrm{Y}$$\overline{\cup \mathcal{F}’}x\cap \mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}=\omega$$\mathrm{Y}$$\overline{A}^{X}$$\mathrm{Y}$$\mathrm{Y}$$X=(\omega+1)^{2}\backslash (\{\omega\}\cross\omega),$$\mathrm{Y}=(\omega+1)\cross\{\omega\}$$\mathrm{Y}$$Y$$\mathrm{Y}$$\mathrm{Y}=\{\omega\}$125

Remark 5.1. In [10], Grabner et.al. defined that is weakly cp-paracompact
in $X$ if for every open cover , there is a closed partial refinement $F$ suchthatis weakly closure preserving with respect to Y. In [12],andGrabner et.al. modified the definition of weak -paracompactness in $X$ asfollows;
is weakly $cp$ -paracompact in $X$ if for every open cover , there isa (not necessarily clos ) partial refinement .7‘ such thatand .7‘ isweakly closure preserving with respect to Y. They commented in [12] that the
new definition of weak cp–paracompactness in $X$ appears to be weaker. Notethat is 2-cpc-paracompact in $X$ if and only if is weakly -paracompact in$X$ (in the sense in [10]). Moreover,is 2-cpparacompact in $X$ if is weakly
cp–paracompact in $X$ (in the sense of revised definition in [12]). Assuming isstrongly regular in $X$ , these notions are quivalent as in Diagram 4.
$\mathrm{Y}$$\mathcal{U}$$\mathrm{Y}\subset\cup F$$\mathcal{F}$$\mathrm{c}\mathrm{p}$$\mathrm{Y}$$\mathcal{U}$$\mathrm{Y}\subset\cup F$$e\mathrm{d}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{c}\mathrm{p}$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{Y}$$e$Remark 5.2. In [11, Lemma 2.2], Grabner et.al. assert that if a closed collec$\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\dot{\mathrm{a}}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{p}.\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}F\mathrm{i}\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{a}e\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}Y\mathrm{a}\mathrm{n}\mathrm{d}A\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{Y}$,$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}A\subset X\backslash ^{\frac{\mathrm{k}1\mathrm{y}\mathrm{c}1\mathrm{o}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{r}}{\cup(\mathcal{F}\backslash \{F\in F|F\cap \mathrm{Y}\neq\emptyset\})}X}.\mathrm{H}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{v}e\mathrm{r},$consider $X=\omega+1,$ $\mathrm{Y}=A=\{\omega\}$ and $F=\{\{n\}|n\in\omega\}$ .

To discuss the notions by Grabner et.al. and our notions defined in Section
2, let us introduce some other notions relative paracompactness. We define that-paracompact, $a’- lfc$ -paracompact) in $X$ if
is -paracompact (respectively,
for every open cover of $X$ there exists an open partial refinement (resp ctively,anda partial refinement, a clos partial refinement) of such thatis locally finite in $X$ .
-paracompact (respectively, -cp-paracompact,
We also say that is-paracompact) in $X$ if for every open cover of $X$ there exists an open
partial refinement (respectively, a partial refinement, a closed partial refinement)
and is closure-preserving in $X$ . Notice that it is easy tosuch that-paracompact in $X$ if and only if
see that a subspace of a space $X$ isis cp–paracompact in $X$ in the sense of Grabner et.al. [10]; this fact is pointed
out in [12] assuming that $X$ is Hausdorff. But, in Proposition 5.3 below, we show
-paracompactness without-paracompactness is coincident withthat
any additional condition.
The notion of -paracompactness is intermediate between and 2-paracompactness, and is independent from 1-paracompactness. It is obvious thatparacompactness is equivalent to -paracompactness for closed subspaces. On
the other hand, there exist a Tychonoff space $X$ and its subspace such thatis -paracompact in $X$ , but not a-paracompact in $X$ (consider $X=\omega+1$
and $Y=\omega$ ) .Moreover, there exist a Tychonoff space $X$ and its subspace
is 1-paracompact in $X$ , but not -paracompact in $X$ (consider
such that.
andIn the rest of this section, we consider the following Proposition 5.3.
$\mathrm{Y}$$\alpha’$$\alpha’- lf$$e$$\mathcal{U}$$V$$e\mathrm{d}$$Y\subset\cup \mathcal{V}$$\mathcal{U}$$\mathcal{V}$$\mathrm{Y}$$\alpha’$$\alpha^{J_{-}}cpo$$\alpha’- cpc$$\mathcal{V}$$\mathcal{U}$$\mathrm{Y}\subset\cup \mathcal{V}$$\mathcal{V}$$Y$$\alpha’- cpc$$\mathrm{Y}$$\alpha’- cpc$$\alpha’- lfc$$\alpha’$$\alpha-$$\alpha’-$$\alpha$$\mathrm{Y}$$\mathrm{Y}$$\alpha’$

$Y$

$.$

$\mathrm{Y}$$X=A(\omega_{1})\cross(\omega+1)\backslash \{\langle\infty,\omega\rangle\}$$\alpha’$$\mathrm{Y}=D(\omega_{1})\cross\omega)$

126

Proposition 5.3. For a subspace of a space $X$ , the following statements areequivalent.
is a’-lfc-paracompact in $X$ .
is a’-cpc-paracompact in $X$ .
is regular.
-paracompact in $X$ andisis regular.
is a’-cp-paracompact in $X$ andis paracompact Hausdorff.
$\mathrm{Y}$$(a)\mathrm{Y}$$(b)\mathrm{Y}$$\overline{\mathrm{Y}}^{X}$$(c)\mathrm{Y}$$\alpha’- lf$$\overline{\mathrm{Y}}^{X}$$(d)\mathrm{Y}$$(e)\overline{\mathrm{Y}}^{X}$

Grabner et.al. [10, Theorem 35] (respectively, [12, Theorem 8]) proved thatand $(e)$ in Proposition 5.3 above are equivalent assuming thatthe statements$X$ is regular (respectively, Hausdorff).
$(b)$Lemma 5.4. Let be a subspace of a space X. Then, the following statement8
are equivalent.
) paracompact in $X$ andis regular.
$(a)Y$ is a’-lf- (respectively,
is a’-lfc- (respectively, $a’- cpc-$) paracompact in $X$ .
) paracompact in $X$ .
is a’-lfc- (respectively,$\mathrm{Y}$$\overline{\mathrm{Y}}^{X}$$a^{J_{-}}cp-$$(b)\overline{Y}^{X}$$\alpha^{J_{-}}cpc-$$(c)\mathrm{Y}$Proposition 5.3 and Lemma 5.4 induce the following.

Corollary 5.5. Assume thatis a’-lf-paracompact in $X$ .

$\overline{\mathrm{Y}}^{X}$

Moreover, by applying Theoris regular. If$e\mathrm{m}3.5$$\mathrm{Y}$is a’-cp-paracompact, then$\mathrm{Y}$, we have

Corollary 5.6. Assume that is closed in $X$ and satisfies the condition $(*)$
in Proposition 3.1. If is a’-cpo-paracompact in $X$ , then is -paracompact
in $X$ .
We conclud this note by the following implications among $a’$ -cases. These
implications directly follow from definitions, Proposition 5.3, Corollaries 5.5 and5.6. Here, the symbol $(*)$ denot es the condition $(*)$ in Proposition 3.1.
$\mathrm{Y}$$\mathrm{Y}$$\alpha’$$\mathrm{Y}$$\mathrm{Y}$

$e$

$Y$

is

$\mathrm{Y}$is$\overline{\mathrm{Y}}^{\mathrm{X}}\in \mathcal{T}_{3}$is-paracompact$\mathrm{Y}$$a’- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}-a’- lf- \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}=a’- lfc$$\mathrm{Y}\in C_{\mathrm{X}}(*)|\downarrow \mathrm{i}\mathrm{n}X$$\mathrm{Y}$

is

$Y$

$\alpha’-cp\not\in \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{t}arrow\alpha’- c_{\Psi}$

in $X$$\mathrm{i}\mathrm{n}_{\mathrm{I}^{\mathrm{x}}}$$\overline{\mathrm{Y}}^{\mathrm{X}}\in \mathcal{T}_{3}|\downarrow \mathrm{i}\mathrm{n}X$is$\overline{\mathrm{Y}}^{\mathrm{X}}\in \mathcal{T}_{3}$-ParacomPact
in $X$DIAGRAM 6

is-paracompact
in $X$$=\alpha’- cpc$$\mathrm{Y}$127

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Nara Women’s University Secondary School,
HigashilCidera, Nara 630-8305, Japan-mail: kawaguchi@cc.nara-wu.ac.jp$E$