@l $f $$c ,sec;n #f.a4 ,prop]ties ,f.d 9 @m ,t3, @l ,p>a-,l9del(1
,nonp>acompact ,spaces4 $p ,0exam9+ ! examples (! previ@rs two *apt]s &
mak+ "s simple modific,ns1 we c 9ve/igate : topological prop]ties c g al;g
) 2+ @m ,t3 @l & p>a-,l9del( b n p>acompact4 ,^! f\r ma9 examples w 2
ref]r$ to1 9 ord]1 z @m ,x(4), ,y(4), ,x(5), @l & @m ,y(5)_4 @l ,e>li]
not,nal def9i;ns / hold1 except ": !y >e specific,y ov]rul$4 $p @m ,x(4)
@l & @m ,y(4)_1 @l ! examples us+ a normal @m .w1-,cantor @l tree1 dep5d
on ! exi/;e ( s* a tree2 = example1 !y c 2 3/ruct$ "u @m ,,ma(.w1) @l b _c
"u ,,ch4 ,9 _! favor1 ^! spaces >e f/ c.ta#4 ,9 fact1 ea* ( !m is
developa# & has a @m .s-_ @l loc,y c.ta# base2 only @m ,x(4) @l has a .s-_
@l 4jo9t base4 ,on ! o!r h&1 @m ,x(5) @l & @m ,y(5)_1 @l ! examples us+
,b+'s ;,g1 >e 8r1l0 examples4 ,b ^! spaces fail 6be f/ c.ta#4 ,9 fact1 !y
bo? h "* @m 2^.w^;1_4 @l ,! two nonnormal spaces1 @m ,x(4) @l & @m ,x(5)_1
@l >e scre5a#2 = ea*1 ! po9t-f9ite op5 ref9e;t 3/ruct$ 9 %[+ metacompact;s
is al @m .s-_ @l 4jo9t 79 fact ! union ( two pairwise 4jo9t collec;ns74 ,!
two normal spaces1 @m ,y(4) @l & @m ,y(5)_1 @l >e n scre5a#2 = ea*1 ! op5
cov] us$ 9 %[+ t ! space is n collec;nwise normal has no @m .s-_ @l 4jo9t
op5 ref9e;t4 ,:ile all f\r spaces >e -pletely regul>1 n"o >e collec;nwise
normal4 ,all >e metacompact & c.tably p>acompact4 $p ,an elegantly simple
te*nique devis$ 0;,f4 ,burton ,j"os uses a space : is regul> b n normal
63/ruct a new space : is regul> b n -pletely regul>4 ,0apply+ ? te*nique
6! spaces @m ,x(4) @l & @m ,x(5)_1 @l we get p>a-,l9del( @m ,t3-spaces @l
: >e n -pletely regul>4 $p $$ub #f.a.a3 ,3/ruc;n $$uf 7,j"os ,7#g7'74 ,let
;,x 2 a topological space : is regul> b n normal4 ,acly1 let ;,h & ;,k 2
4jo9t clos$ subsets ( ;,x : _c 2 sep>at$ 04jo9t op5 sets4 ,let ;,n 2 ! set
( natural numb]s @m .(1, 2, '''.) @l )! 4crete topology4 ,let @m ,z' @l 2
! id5tific,n space @m ,x @* ,n_/@: @l ": ! equival;e rel,n @m @: @l on @m
,x @* ,n @l is g5]at$ 0! rule @m (x, n) @: (x, n+1) @l if3 @m x @e ,h @l &
;n is ev51 or @m x @e ,k @l & ;n is odd4 ,f9,y1 let @m ,z .k ,z' .+ .(p.)
@l ": @m p /@e ,z'_4 @l ,= ea* ;n 9 ;,n1 let @m ,x;n .k .((x, n)\ x @e
,x.)_/@:_1 @l : is homeomorphic to ;,x4 ,6topologize ;,z1 @m let ,u _"k
,z' @l 2 op5 9 ;,z iff ;,u is op5 9 @m ,z'_1 @l or equival5tly iff @m ,u
.% ,x;n @l is op5 9 @m ,x;n @l = ea* ;n 9 ;,n4 ,al1 if
 
@l $p ,:ile ;,z is regul>1 ,j"os %[s t ;,z is n -pletely regul>4 ,=
example1 "! c 2 no 3t9u\s func;n : is #j at ;p & #a at e po9t (! clos$ set
@m ,x1_4 @l ,? 3/ruc;n pres]ves _m topological prop]ties1 9clud+ @m ,t1-s
@l & p>a-,l9del(;s4 $p $$ub #f.a.b3 ,example $$uf @m (,,ma(.w1))_4 @l ,let
;,x 2 @m ,x(4)_1 @l & use ;,x 9 ,3/ruc;n #f.a.a4 ,!n ;,z is a p>a-,l9del(
,moore space : is n -pletely regul>4 ,l ;,x1 ;,z has a @m .s-_ @l 4jo9t1
@m .s-_ @l loc,y c.ta# base & is metacompact & c.tably p>acompact4 $p $$ub
#f.a.c3 ,example4 $$uf ,let ;,x 2 @m ,x(5)_1 @l & uild ;,z 0apply+
,3/ruc;n #f.a.a to ;,x4 ,!n ;,z is @m ,t3 @l & p>a-,l9del( b n -pletely
regul>4 ,l ;,x1 ;,z has "* @m 2^.w^;1 @l & is metacompact & c.tably
p>acompact4 @m $4 @l $p ,we c al get @m ,t3, @l p>a-,l9del( spaces : >e n
metacompact 0modify+1 9 turn1 @m ,y(4) @l & @m ,y(5)_4 @l ,we replace sets
( two 9t]lac+ func;ns 0sets ( @m .w-many @l mutu,y 9t]lac+ func;ns4 $p
$$ub #f.a.d3 ,example $$uf @m (,,ma(.w1))_4 @l ,= ea* @m n .1: #1_1 let
,j;n .k .(.(.s;i"\ i @e .w.) _"k ,p;n"\ (i) .s;i"(0) /.k .s;j"(0) :5 i /.k
j, (ii) @l ! func;ns @m .s;i @l >e mutu,y 9t]lac+1 & 7iii7 ! sequ;es @m
s;m"(.s;i"\ m)@rn @l >e ! same = e @m i @e .w.); let ,j;n^@# .k " .+%m_
.1:_ n] ,j;m_4 @l ,n[ let @m ,j .k ,j;1^@# @l & @m ,y .k ,f .+ ,j_4 @l $p
,n[ we topologize ;,y4 ,! po9ts ( ;,j >i ,= ea* @m n .1: #1 @l & ;f 9 ;,f1
! set @m ,u;n"(f) .k @(f@rn_3 ,f@) .+ .(.(.s;i"\ i @e .w.) @e ,j;n^@#"\
.s;i"@rn .k f@rn @l = "s @m i @e .w.) @l is op54 ,! s+letons on ;,j tgr )
^! op5 sets provide a base =! topology on ;,y4 $p ,! space ;,y has all !
prop]ties m5;n$ = @m ,y(4) @l except = metacompact;s4 ,prov+ t ;,y is n
metacompact is v simil> 6prov+ t @m ,y(4) @l is n collec;nwise normal4
,"h1 ag1 we modify ! >gu;t %[+ t @m ,x(4) @l is n normal4 ,= ea* @m .a "k
.w1_1 let ,f;.a .k .(f@e,f\ f(0) .k .a.) @l & @m let ,u;.a @l 2 an op5 set
) @m ,u;.a .% ,f .k ,f;.a_4 @l ,i claim t ! op5 cov] @m @;,u .k .(,u;.a"\
.a "k .w1.) .+ .(,j.) @l has no po9t-f9ite op5 ref9e;t4 ,6see ?1 let @m
@;,v @l 2 an >bitr>y op5 ref9e;t = @m @;,u_4 @l ,) respect 6! op5 cov] @m
@;,v_1 @l def9e @m c(f) @l = ea* ;f 9 @m ;,f, ,a;n @l = ea* @m n .1: #1,
@l & ;,a z 9 ! pro( t @m ,x(4) @l is n normal4 ,al let @m n .1: #1, ;,b,
;,m, @l & ;,c 2 z 9 t pro(4 ,n[1 0,lemma #c.b.i1 *oose mutu,y 9t]lac+
memb]s ( @m ;,c_1 .s;i @l = e @m i @e .w_1 @l no two ( : agree at #j4 ,let
;y 2 ! po9t @m .(.s;i"\ i @e .w.)_4 @l ,!n1 = ea* @m i @e .w_1 st(y, @;,v)
meets ,f;.s;;i"(0)_4 @l ,b ea* memb] ( @m @;,v @l meets at mo/ "o @m
,f;.a_4 @l ,s9ce ;y ?us lies 9 9f9itely _m memb]s ( @m @;,v_1 @;,v @l is n
po9t-f9ite4 ,"!=e ;,y is n metacompact4 @m $4
 
@l $p $$ub #f.a.e3 ,example4 $$uf ,= ea* @m n .1: #1_1 let ,j;n .k
.((.(.s;i"\ i @e .w.), r) @l ": @m .(.s;i"\ i @e .w.) _"k ,p;n @l & @m r
@e ,t;n"\ (i) .s;i"(0) /.k .s;j"(0) @l :5 @m i /.k j, @l & 7ii7 ! func;ns
@m .s;i @l >e mutu,y 9t]lac+.); let @m ,j;n^@# .k " .+%m_ .1:_ n] ,j;m_4
@l ,n[ let @m ,j .k ,j;1^@# @l & @m ,y .k ,f .+ ,j_4 @l $p ,n[ we
topologize ;,y4 ,! po9ts ( ;,j >e isolat$4 ,= ea* @m n .1: #1, ;f @l 9
;,f, & ;t 9 @m ,t;n_1 @l ! set @m ,u;n"(f, t) .k @(f@rn_3 ,f@) .+
.((.(.s;i"\ i @e .w.), r) @e ,j;n^@#"\ (i) .s;i"@rn .k f@rn @l = "s @m i
@e .w, @l & 7ii7 giv5 any @m ,e @e t .% ,s;i @l = "s @m i "k: n_1 ,e @e r
iff f@ri @e ,e.) @l is op54 ,! s+letons on ;,j tgr ) ^! op5 sets provide a
base =! topology on ;,y4 $p ,! space ;,y has all (! prop]ties m5;n$ = @m
,y(5) @l except = metacompact;s4 ,%[+ t ;,y is n metacompact is analog\s
6%[+ ! same result 9 ,example #f.a.d4 ,a pro(1 l ! pro( %[+ t @m ,y(5) @l
is n collec;nwise normal1 is bas$ on ! >gu;t %[+ t @m ,x(5) @l is n
normal4 @m $4 @l $p ,0tak+ 4jo9t unions ( "s (! examples ( p>a-,l9del( @m
,t3-spaces @l alr giv51 we c get new "os : -b9e _! negative prop]ties4 ,we
h se5 @m ,t1, @l p>a-,l9del( spaces : >e regul> b n -pletely regul>1
-pletely regul> b n normal1 or normal b n collec;nwise normal4 ,we h j se5
two examples : elim9ate metacompact;s f ! normal spaces4 ,0tak+ a 4jo9t
union ) "o ( ^! two spaces1 we c elim9ate metacompact;s f any (! nonnormal
spaces z well4 ,h["e1 ) ? modific,n we al elim9ate scre5abil;y2 9 fact1 if
a space is scre5a# & c.tably p>acompact1 !n x is metacompact4 ,simil>ly1
we c elim9ate j scre5abil;y f ! nonnormal spaces 0tak+ a 4jo9t union ) an
appropriate "o (! normal spaces4 $p ,if we >e will+ 6give up
developabil;y1 we c 7"u @m ,,ma(.w1)) @l elim9ate @m .s-_ @l loc,y c.ta#
bases :ile reta9+ f/ c.tabil;y4 ,0a !orem ( ;,v4 ,f$orcuk ,7#aj7'1 e
p>acompact @m ,t3-space @l )a @m .s-_ @l loc,y c.ta# base is metriza#4 ,s
any p>acompact1 nonmetriza# @m ,t3-space_1 @l = example ! ,sorg5frey l9e
;,e1 is @m ,t3 @l p>a-,l9del( b has no @m .s-_ @l loc,y c.ta# base 7& h;e
no develop;t1 z e p>a-,l9del( ,moore space has a @m .s-_ @l loc,y c.ta#
base74 ,tak+ ! 4jo9t union ( ei @m ,x(4) or ,y(4) @l 7"u @m ,,ma(.w1)) @l
) ;,e w l1ve a f/ c.ta#1 p>a-,l9del(1 nonp>acompact ,t3-space : has no @m
.s-_ @l loc,y c.ta# base & is n a ,moore space4
 
@l $p ,0a !orem ( ;,e4 ,mi*ael ,7#h7'1 a 3t9u\s1 clos$ image (a p>acompact
@m ,t2-space @l is xf p>acompact4 ,h["e1 9 ! next example we see t a
,moore space : is a 3t9u\s1 clos$ image (a p>a-,l9del( ,moore space ne$ n
2 p>a-,l9del(4 $p $$ub #f.a.f3 ,example4 $$uf ,let ;,x 2 @m ,x(5)_4 @l
,us+ ! not,n ( ,example #e.a.d1 def9e an equival;e rel,n @m @: @l on ;,x
as foll[s4 ,def9e @m @: @l on ;,i 0lett+ @m (.s, .t, r) @: (.s', .t', r')
iff .s' .k .s, .t' .k .t, @l & @m r .% ,s1 .k r' .% ,s1_4 @l ,ext5d @m @:
@l 6an equival;e rel,n on ;,x 0al lett+ @m x @: x @l = @m x @e ,h .+ ,k_4
@l ,!n ! projec;n map @m p_3 ,x $o ,x_/@: @l is a 3t9u\s1 clos$ map4 ,al?
! image @m ,t3-space ,x_/@:_1 @l is a ,moore space1 l ! space 9 ,7#a7'1 x
is n p>a-,l9del(4 @m $4 @l $f $$c ,sec;n #f.a4 ,op5 ,"qs4 $p ,2l >e "s
9t]e/+ op5 "qs d1l+ ) p>a-,l9del(1 nonp>acompact @m ,t3-spaces_4 $l @l #a4
,)\t assum+ any extra set-!oretic axioms1 c "o 3/ruct s* a space : is f/
c.ta#8 @m $4 @l $l #b4 ,is "! any s* space : is n c.tably p>acompact8 @m
$4 @l $l #c4 ,is "! any s* space : is collec;nwise normal8 7,diana ,pike
,pal5z ,7#i7' has %[n t e p>a-,l9del(1 monotonic,y normal space is
p>acompact4 ,%e al %[$ t e monotonic,y normal space )a @m .s-_ @l loc,y
c.ta# base is metriza#1 an ext5.n ( ,f$oruk's !orem47 @m $4