@l $f $$c ,sec;n #e.a4 ,a ,p>a-,l9del(1 ,nonnormal @m ,t3-,space @l $p ,9
? *apt] we 3/ruct a space ;,x l t ( ,sec;n #d.a4 ,=! two examples 9 ?
*apt]1 we use ,b+'s example ;,g z a ref];e space1 j z we us$ a normal @m
.w1-,cantor @l tree 9 ! *apt] #d4 ,)\t any extra axioms ( set !ory1 ,b+'s
;,g is a @m ,t4 @l space : is n collec;nwise ,hausdorff4 ,"!=e ! 3/ruc;ns
9 ? *apt] d n require any special set-!oretic assump;ns4 ,h["e1 l ,b+'s
;,g1 ! examples 9 ? *apt] h "* @m 2^.w^;1_4 @l $p $$ub #e.a.a3 ,def9i;n (
,b+'s ;,g $$uf ,7#c7'4 ,let @m ,s .k @;,p(.w1) @l & @m ,g .k ^,s"2_1 @l !
set ( all func;ns f ;,s 9to @m .(0, 1.)_4 @l ,= any @m .a "k .w1_1 let
g;.a_3 ,s $o #2 @l 2 ! "*i/ic func;n = @m .a_2 @l t is1 = any @m ,a _"k
.w1_1 g;.a"(,a) .k @l 1 iff @m .a @e ,a_4 @l ,let @m ,g' .k .(g;.a"\ .a "k
.w1.)_4 @l ,6topologize @m ;,g_1 let all @l ! po9ts ( ;,g-;,g' 2 isolat$4
,al1 giv5 any @m ,a _"k .w1 @l & @m e .k #0 or #1_1 @l ! set @m ,u;,a[e .k
.(g@e,g\ g(,a) .k e.) @l is op54 ,! s+letons on ;,g-;,g' tgr ) all ! op5
sets @m ,u;,a[e @l provide a subbase =! topology on ;,g4 @m $4 @l $p $$ub
#e.a.b3 ,!orem $$uf ,7#c7'4 ,b+'s ;,g is @m ,t4 @l b n collec;nwise
,hausdorff2 no unc.ta# subset (! 4crete set @m ,g' @l c 2 sep>at$ 04jo9t
op5 sets4 @m $4 @l $p $$ub #e.a.c3 ,not,nal ,def9i;n4 $$uf ,giv5 any @m n
.1: #1_1 let ,s;n .k @;,p(,p;n")_1 @l & let @m ,t;n @l 2 ! set ( all f9ite
subsets ( @m " .+%i_ "k:_ n] ,s;i_4 ,let ;,f, ;,h, @l & ;,k 2 z 9 ,*apt]
#d4 ,al let @m @(.r_3 ,f@), @(.r_3 ,h@), @l & @m @(.r_3 ,k@) @l 2 z 9
,*apt] #d1 ": @m .r @e ^."k.w".w1_4 $4 @l $p $$ub #e.a.d3 ,example4 $$uf
,= ea* @m n .1: #1_1 let ,i;n .k .((.s, .t, r) @e ,p;n^2 @* ,t;n"\ (i)
.s(0) /.k .t(0), @l & 7ii7 @m .s @l & @m .t interlace.); let ,i;n^@# .k "
.+%m_ .1:_ n] ,i;m_4 @l ,n[ let @m ,i .k ,i1^@# @l & @m ,x .k ,h .+ ,k .+
,i_4 @l $p ,n[ we topologize ;,x4 ,! po9ts ( ;,i >e isolat$4 ,= ea* @m n
.1: #1, ;h @l 9 @m ;,h, @l & ;t 9 @m ,t;n_1 let ,u;n"(h, t) .k @(h@rn_3
,h@) .+ .((.s, .t, r) @e ,i;n^@#"\ (i) .s@rn .k h@rn, @l & 7ii7 giv5 any
@m ,e @e t .% ,s;i @l = "s @m i "k: n_1 ,e @e r iff h@ri @e ,e.); @l ^!
sets >e op54 ,simil>ly1 = ea* @m n .1: #1, ;k @l 9 @m ;,k, @l & ;t 9 @m
,t;n_1 let ,u;n"(k, t) .k @(k@rn_3 ,k@) .+ .((.s, .t, r) @e ,i;n^@#"\ (i')
.t@rn .k k@rn, @l & 7ii'7 giv5 any @m ,e @e t .% ,s;i @l = "s @m i "k: n_1
,e @e r iff k@ri @e ,e.); @l ^! sets too >e op54 ,! s+letons on ;,i tgr )
^! op5 sets make up ! base @m @;,b @l = ;,x4 ,if @m m "k: n_1 @l notice t
! 9t]sec;n (! basic op5 sets @m ,u;m"(x, t) @l & @m ,u;n"(x, t') is
,u;n"(x, t .+ t'), @l & @m ,u;m"(x, t) @l 3ta9s @m ,u;n"(x, t') if t _"k
t'_4
 
@l $p ,! topological space ;,x has ! foll[+ prop]ties3 $l 7a7 ;,x is @m
,t1 @l & z]o-dim5.nal & h;e al regul> & fur!rmore -pletely regul>4 $l 7b7
;,x has "* @m 2^.w^;1; @l "!=e ;,x is n a ,moore space & does n h a @m
.s-_ @l 4jo9t or a @m .s-_ @l loc,y c.ta# base4 $l 7c7 ;,x is p>a-,l9del(4
$l 7d7 ;,x is n normal4 $l 7e7 ;,x is metacompact4 $l 7f7 ;,x is c.tably
p>acompact4 $p ,9 ! 4cus.n ( ? example1 @m ,x(4) @l w ref] 6! space ;,x (
,example #d.a.e4 ,z 9 ! 4cus.n ( @m ,x(4)_1 @l (t51 )\t fur!r m5;n1 two
correspond+ v]ific,ns 9volv+ po9ts ( ;,h &( ;,k1 respectively1 w 2
repres5t$ 0j "o ( ^!4 $p 7a73 ;,x is @m ,t1 @l exactly z @m ,x(4) @l is2 j
replace any basic op5 set @m ,u;n"(x) @l 9 @m ,x(4) @l 0! basic op5 set @m
,u;n"(x, .f) @l 9 ;,x4 ,6see t ;,x is z]o-dim5.nal1 we v]ify t any basic
op5 set @m ,u .k ,u;n"(h, t)_1 @l ": @m h @e ,h @l & @m .(h@r1.) @e t_1 @l
is clos$4 ,! po9ts ( @m ,i-,u @l &( @m ,h-,u @l c 2 sep>at$ f ;,u j z 9 @m
,x(4) @l -- us+ @m ,u;m"(h', .f) @l 9/1d ( @m ,u;m"(h') @l = @m h' @l 9 @m
,h-,u_4 @l ,9 sep>at+ ;k 9 ;,k f ;,u1 "! >e two possibilities4 ,if @m k(0)
.k h(0)_1 @l !n @m ,u1(k, .f) .% ,u @l is empty -- ag correspond+ 6!
situ,n 9 @m ,x(4)_4 @l ,o!rwise1 s9ce @m k@r1 /.k h@r1_1 ,u1(k, .(h@r1.))
.% ,u @l is empty4 ,"!=e ;,u is clos$4 ,let @m @;,b' @l 3si/ ( all
s+letons on ;,i tgr ) ^? memb]s @m ,u;n"(x, t) @l ( @m @;,b @l ": @m
.(x@r1.) @e t_4 @l ,!n @m @;,b' is also a base @l = ;,x & is clop54 $p
,s9ce ;,x is @m ,t1 @l & z]o-dim5.nal1 x is @m ,t3 @l & fur!rmore
,ty*on(f4 $p 7b73 ;,x has "* @m 2^.w^;1 @l at ea* po9t ( @m ,h .+ ,k @l
s9ce t is ! size ( ea* @m ,s;n_4 @l ,al? ;,x "!=e is n a ,moore space &
does n h a @m .s-_ @l 4jo9t or a @m .s-_ @l loc,y c.ta# base1 @l x is
use;l 6reflect :at c.ta# /ructure ;,x has4 ,= ea* @m n .1: #1_1 let
@;,g;n^0 .k .(,u;n"(h, .f)\ h @e ,h.), @;,g;n^1 .k .(,u;n"(k, .f)\ k @e
,k.), @;,g;n^2 .k .(.(x.)\ x @e ,i - .+ (@;,g;n^0 .+ @;,g;n^1").), @l & @m
@;,g;n .k @;,g;n^0 .+ @;,g;n^1 .+ @;,g;n^2_4 @l ,ea* collec;n @m @;,g;n^e
@l is pairwise 4jo9t = e @m .k #0, #1 or #2_4 @l ,z 9 @m ,x(4)_1 @l ea* @m
@;,g;n @l is loc,y c.ta#1 z @m @;,g;n+1 @l te/ifies4 ,b 9 ? example1 @m "
.+%n_ .1:_ #1] @;,g;n @l is n a base = ;,x4 $p 7c73 ,%[+ t ;,x is
p>a-,l9del( resem#s %[+ t @m ,x(4) @l is p>a-,l9del(4 ,let @m @;,v @l 2 an
op5 cov] ( ;,x4 ,= ea* ;f 9 ;,f_1 let @m c(f) @l 2 ! smalle/ 9teg] ;n s* t
bo? @m ,u;n"(f_<0, t) @l & @m ,u;n"(f_<1, t) @l >e 3ta9$ 9 memb]s ( @m
@;,v @l = "s ;t 9 @m ,t;n_4 @l ,= ea* @m n .1: #1_1 let ,a;n .k .(f@rn\
c(f) .k n.); @l = ea* @m .r @l 9 @m ,a;n_1 @l *oose t;.r @l 9 @m ,t;n @l s
t bo? @m ,u;n"(f_<0, t;.r") @l & @m ,u;n"(f_<1, t;.r") @l ref9e memb]s (
@m @;,v @l = "s 7e7 ;f 9 @m @(.r_3 ,f@)_4 @l ,= 3v5i;e1 = ea* ;x 9 @m ,h
.+ ,k_1 let c(x) .k c(x@r.w) @l & @m t(x) .k t;.r_1 @l ": @m .r .k
x@rc(x)_4 @l ,al1 = e ;x 9 @m ,h .+ ,k_1 let t'(x) .k t(x) .+ .(,a;i"\ #1
"k: i "k: c(x).)_4
 
@l $p ,= ea* ;x 9 @m ,h .+ ,k_1 let ,w(x) @l 2 ! op5 set @m ,u;c(x)"(x,
t'(x))_4 @l ,!n @m h@rc(h) @l det]m9es @m ,w(h) @l = ;h 9 ;,h2 simil>ly1
@m k@rc(k) @l det]m9es @m ,w(k) @l = ;k 9 ;,k4 ,n[ let @m @;,w^0 .k
.(,w(h)\ h @e ,h.), @;,w^1 .k .(,w(k)\ k @e ,k.), @l & @m @;,w^2 .k
.(.(x.)\ x @e ,i- .+ (@;,w^0 .+ @;,w^1").)_4 ,let @;,w @l 2 ! op5 cov] @m
@;,w^0 .+ @;,w^1 .+ @;,w^2_1 @l ref9+ @m @;,v_4 @l $p ,! possibilities =
9t]sec;n 2t memb]s ( @m @;,w @l %[ a picture l t 9 @m ,x(4)_4 @l ,bo? @m
@;,w^0 .+ @;,w^2 @l & @m @;,w^1 .+ @;,w^2 @l >e pairwise 4jo9t collec;ns4
,n[ suppose t @m ,w(h) meets ,w(k) @l ": @m h @e ,h @l & @m k @e ,k_4 @l
,6see t @m c(h) .k c(k)_1 @l suppose ? is n true4 ,)\t loss ( g5]al;y
assume t @m c(h) "k c(k)_1 @l & let @m n .k c(h)_4 @l ,*oose @m (.s, .t,
r) @l 9 @m ,w(h) .% ,w(k)_4 @l ,?us @m ,a;n @e r s9ce h@rn @e ,a;n_4 @l ,b
@m ,a;n /@e r @l s9ce @m k@rn /@e ,a;n_4 @l ,? is ( c\rse impossi#1 & s @m
c(h) .k c(k)_4 @l $p ,n[ we v]ify t @m @;,w @l is loc,y c.ta#4 ,= any ;x 9
@m ;,i1 @m .(x.) @l meets at mo/ two memb]s ( @m @;,w_4 @l ,= e ;x 9 @m ,h
.+ ,k_1 let ,z(x) @l 2 ! op5 set @m ,w(x) .% ,u;c(x)+1"(x, .f)_4 @l ,6*eck
t ea* @m ,z(x) @l avoids all b c.tably _m memb]s ( @m @;,w_1 @l 3sid] @m
,z(h) @l ": @m h @e ,h @l & @m c(h) .k n_4 ,z(h) @l meets j "o memb] ( @m
@;,w^0 .+ @;,w^2 @l -- "nly1 @m ,w(h)_4 @l ,2c @m ,z(h) @l ref9es @m
,w(h)_1 @l e memb] ( @m @;,w^1 cet by @m ,z(h) lies @l 9 @m @;,w;n^1 @l --
": @m @;,w;n^1 .k .(,w(k)\ k @e ,k @l & @m c(k) .k n.) @l -- & is ?us
3ta9$ 9 a memb] ( @m @;,g;n^1_4 @l ,b s9ce @m ,z(h) @l al ref9es @m
,u;n+1"(h, .f)_1 @l x meets only c.tably _m memb]s ( @m @;,g;n^1_1 @l ea*
( : 3ta9s at mo/ "o memb] ( @m @;,w;n^1_4 @l ,?us @m ,z(h) @l meets only
c.tably _m memb]s ( @m @;,w^1 @l & h;e ( @m @;,w_4 @l ,s @m @;,w @l is
loc,y c.ta#1 & "!=e ;,x is p>a-,l9del(4 $p 7d73 ,6v]ify t ;,x is n normal1
let @m ;,v1 @l & @m ;,v2 @l 2 >bitr>y op5 sets : 3ta9 ! 4jo9t clos$ sets
;,h & ;,k1 respectively4 ,z 9 @m ,x(4)_1 @l we w see t @m ,v1 @l & @m ,v2
@l 9t]sect & ?us t ;,h & ;,k _c 2 sep>at$ 04jo9t op5 sets4 ,) @m @;,v @l !
op5 cov] @m .(,v1, ,v2, ,i.)_1 let c(f) @l = ea* ;f 9 @m ;,f, ,a;n @l =
ea* @m n .1: #1, @l & @m t;.r @l = ea* @m .r @l 9 "s @m ,a;n @l 2 z 9 !
pro( ( 7c74 ,let @m ,a .k " .+%n_ .1:_ #1] ,a;n_4 @l ,s9ce e ;f 9 ;,f has
a re/ric;n 9 ;,a1 0,lemma #c.b.aa1 *oose @m n .1: #1 @l &a subset ;,b ( @m
,a;n @l s t ;,b is ;n-full4
 
@l ,n[1 0,lemma #c.b.ab1 f9d a subset @m ,b' @l ( ;,b s t ! family @m
(t;.r_3 .r ,an ;n-full .,d-system @l ) root family @m (t;.?_3 .? @e
pr,b')_4 @l ,= ea* @m .r @l 9 @m ,b'_1 let r;.r .k " .+%i_ "k:_ n] .(,e @e
t;.r .% ,s;i"\ .r@ri @e ,e.)_1 @l ! set ( ^? memb]s ( @m t;.r @l on : @m
.r @l votes yes4 ,def9e ! 8mat*+0 func;n ,m3 @m ,b' $o @;,p(t;.f") @l
0lett+ @m ,m(.r) .k r;.r .% t;.f_1 @l ! set ( ^? memb]s ( @m t;.f @l on :
@m .r @l votes yes1 = ea* @m .r @l 9 @m ,b'_4 @l ,s9ce "! >e only f9itely
_m possibilities = @m ,m(.r)_1 @l 0,lemma #c.b.aj1 let ;,c 2 an ;n-full
subset ( @m ,b' @l on : ;,m is 3/ant2 ?us e @m .r @l 9 ;,c votes ! same
way on ! memb]s ( @m t;.f_4 @l ,f9,y1 0,lemma #c.b.i1 f9d two 9t]lac+
memb]s @m .s @l & @m .t @l ( ;,c ) @m .s(0) /.k .t(0)_4 @l ,let @m r .k
r;.s .+ r;.t_4 @l ,9 v]ify+ t ! po9t @m (.s, .t, r) @e ,v1 .% ,v2_1 @l
3sid] ;,e 9 @m t;.s_4 @l ,suppose @m ,e @e r;.t_4 @l ,!n @m ,e @e t;.s .%
r;.t_1 @l a subset ( @m t;.s .% t;.t .k t;.f_4 @l ,s @m ,e @e r;.t .% t;.f
.k r;.s .% t;.f_4 @l ,?us @m ,e @e r iff ,e @e r;.s iff .s @l votes yes on
;,e4 ,simil>ly1 giv5 ;,e 9 @m t;.t_1 ,e @e r iff ,e @e r;.t iff .t votes
yes on ;,e_4 @l ,s1 = any ;h 9 @m @(.s_3 ,h@) @l & ;k 9 @m @(.t_3 ,k@)_1
(.s, @m .t, r) @l 2l;gs to @m ,u;n"(h, t;.s") .% ,u;n"(k, t;.t") @l & 9
turn to @m ,v1 .% ,v2_4 @l ,"!=e ;,x is n normal4 $p 7e73 ,we n[ use !
pro( ( 7c7 6%[ t ;,x is metacompact1 z 9 ,sec;n #d.a4 ,giv5 any op5 cov]
@m @;,v @l ( ;,x1 ! loc,y c.ta# op5 ref9e;t @m @;,w @l 3/ruct$ 9 prov+ 7c7
is al po9t-f9ite4 ,b m simply1 let ! func;n ;c on @m ,f .+ ,h .+ ,k, @l
ea* @m ,a;n, @l ea* @m t;.r, @l &! func;n ;t on @m ,h .+ ,k @l 2 z 9 !
pro( ( 7c74 ,!n let @m @;,u^0 .k .(,u;c(h)"(h, t(h)\ h @e ,h.) @l & @m
@;,u^1 .k .(,u;c(k)"(k, t(k))\ k @e ,k.)_4 @l ,def9e @m @;,u^2 @l & @m
@;,u @l acly z 9 ! pro( t @m ,x(4) @l is metacompact1 & 3t9ue z 9 t pro(4
,notice t we cd j z well h def9$ ! func;ns ;c & ;t 9dep5d5tly on ;,h &
;,k4 $p 7f73 ,we al use ! pro( ( 7c7 6%[ t ;,x is c.tably p>acompact1 z 9
,sec;n #d.a4 ,=a c.tably 9f9ite op5 cov] @m @;,v .k .(,v;i"\ i .1: #1.) @l
( ;,x1 = ea* ;f 9 ;,f let @m c(f) @l 2 ! smalle/ 9teg] ;n s* t bo? @m
,u;n"(f_<0, t) @l & @m ,u;n"(f_<1, t) @l >e 3ta9$ 9 sets @m ,v;i @l ) @m i
"k: n @l = "s ;t 9 @m ,t;n_4 @l ,def9e ea* @m ,a;n; @l ea* @m t;.r; @l &
@m c(x), t(x), @l & @m ,w(x) @l = ea* ;x 9 @m ,h .+ ,k @l acly z 9 ! pro(
( 7c74 ,n[ 3t9ue z 9 ! pro( t @m ,x(4) @l is c.tably p>acompact4 @m $4 @l
$p ,9 ? example1 ! memb]s (! sets @m ,t;n @l >e ess5tial 9 gett+ ;,x 6be
p>a-,l9del( & al c.tably p>acompact4 ,= ea* @m i .1: #1_1 @l id5tify @m
,p;i @l "! set @m ,g' @l ( "*i/ic func;ns 9 ,b+'s example ;,g 0a "o-"o1
onto func;n @m s;i; ,g' @l plays ! same role "h t ,t'1 ! top level (! @m
.w1-,cantor @l tree ;,t1 plays 9 ! example ;,x ( ,*apt] #d4
 
@l $p ,let ;h 9 ;,h & @m n .1: #1 @l 2 giv54 ,any 8;nth level0 op5
nei<borhood ( ;h -- "o ^: 9t]sec;n ) @m ,h .+ ,k is @(h@rn_3 ,h@) @l & ^:
9t]sec;n ) ;,i is det]m9$ 0xs 9t]sec;n ) @m ,i;n @l -- c 2 ?9n$ d[n 0an
associ,n )! op5 nei<borhoods ( @m s;i"(h@ri) @l 9 ;,g = ea* @m i "k: n_4
@l ,m precisely1 giv5 ;t 9 @m ,t;n_1 @l suppose t @m t .% ,s;i .k t;i @l =
ea* @m i "k: n_4 @l ,= ea* @m i "k: n_1 @l ! ;nth level op5 nei<borhood @m
,u;n"(h, t) @l ( ;h is associat$ )! op5 nei<borhood @m .% .(,u;,e[0"\ ,e
@e t;i @l & @m h@ri /@e ,e.) .% .% .(,u;,e[1"\ ,e @e t;i @l & @m h@ri @e
,e.) @l ( @m h@ri @l 9 ;,g2 we take ! 9t]sec;n ( an empty collec;n 6be !
5tire space ;,g4 ,simil>ly1 giv5 ;k 9 ;,k & @m n .1: #1_1 @l any ;nth
level op5 nei<borhood ( ;k c 2 ?9n$ d[n 0an associ,n )! op5 nei<borhoods (
@m s;i"(k@ri) @l 9 ;,g = ea* @m i "k: n_4 @l ,if ! nei<borhood 9 ;,g
associat$ ) @m h@ri @l & @m t .% ,s;i @l is 4jo9t f t associat$ ) @m k@ri
@l & @m t' .% ,s;i @l = "s @m i "k: n_1 @l !n @m ,u;n"(h, t;i") @l & @m
,u;n"(k, t') @l >e 4jo9t4 $p ,s9ce ! space ;,g is normal1 ? ?9n+ d[n
provides 5 sep>,n 6make ;,x p>a-,l9del(4 ,=1 z 9 ,*apt] #d1 x provides a
way ( 3troll+ ! 8levels0 ( 9t]sect+ op5t! o!r h&1 ;,g is n collec;nwise
,hausdorff s9ce no unc.ta# subset (! 4crete set @m ,g' @l c 2 sep>at$
04jo9t op5 sets4 ,"!=e a collec;n ( memb]s @m t;.r @l ( @m ,t;n @l _c
4t+ui% 2t unc.tably _m func;ns @m .r @l 9 any giv5 @m ,p;n_4 @l ,s ! ?9n+
d[n does n provide 5 sep>,n 6make ;,x normal4 $p ,9 ? sec;n we 3/ruct a
space ;,x l t ( ,sec;n #d.a4 ,=! two examples 9 ? *apt]1 we use ,b+'s ;,g
z a ref];e space1 j z we us$ a normal @m .w1-,cantor @l tree 9 ,*apt] #d4
,)\t any extra axioms = set !ory1 ,b+'s ;,g is a @m ,t4 @l space : is n
collec;nwise ,hausdorff4 ,"!=e ! 3/ruc;ns 9 ? *apt] d n require any
special set-!oretic assump;ns4 ,h["e1 l ,b+'s ;,g1 bo? examples 9 ? *apt]
h "* @m 2^.w^;1_4 @l $p ,n[ we use ! te*niques (! previ\s sec;n 6build a
topological space l ! ssace ;,y ( ,sec;n #d.a4 ,once ag1 ? does n require
any extra set-!oretic assump;ns4 $p #a3 ,example4 ,giv5 any @m n .1: #1_1
let ,j;n .k .((.(.s, .t.), r) @l ": @m .(.s, .t.) _"k ,p;n @l & @m r @e
,t;n"\ (i) .s(0) /.k .t(0), @l & @m (ii) .s @l & @m .t interlace.)_4 ,let
,j;n^@# .k " .+%m_ .1:_ n] ,j;m_4 ,let ,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 let ,u;n"(f, t) .k @(f@rn_3 ,f@) .+
.((.(.s, .t.), r) @e ,j;n^@#"\ (i) .s@rn .k f@rn or .t@rn .k f@rn, @l &
7ii7 giv5 any @m ,e @e t .% ,s;i @l = "s @m i
 
@l $p ,! topological space ;,y has ! foll[+ prop]ties3 $l 7a7 ;,y is @m
,t1 @l & z]o-dim5.nal & al normal4 $l 7b7 ;,y has "* @m 2^.w^;1; @l "!=e
;,y is n a ,moore space & does n a @m .s-_ @l 4jo9t or a @m .s-_ @l loc,y
c.ta# basis4 $l 7c7 ;,y is p>a-,l9del(4 $l 7d7 ;,y is n collec;nwise
normal4 $l 7e7 ;,y is metacompact4 $l 7f7 ;,y is c.tably p>acompact4 $p ,9
! 4cus.n ( ? example1 ! topological space ;,x ref]r$ to w 2 t ( ,example
#e.a.d4 ,al1 @m ,y(4) @l w ref] 6! space ( ,example #d.b.a4 $p 7a73 ;,y is
@m ,t1 @l exactly z @m ,y(4) @l is4 ,%[+ t ;,y is z]o-dim5.nal is v simil>
6%[+ ! same result = ;,x4 ,3sid] @m ,u .k ,u;n"(f, t) @l ": @m n .1: #1, f
@e ,f, t @e ,t;n, @l & @m .(f@r1.) @e t; @l we v]ify t ;,u is clos$4 ,giv5
;y 9 @m ,j-,u_1 .(y.) @l sep>ates ;y f ;,u z 9 @m ,y(4)_4 @l ,9 sep>at+ @m
f' @l 9 @m ,f-,u @l f @m ;,u_1 @l "! >e two possibilities4 ,if @m f'(0) .k
f(0)_1 @l sep>ate @m f' @l f ;,u z 9 @m ,y(4) @l -- us+ @m ,u;m"(f', .f)
@l 9/1d ( @m ,u;m"(f')_4 @l ,o!rwise1 s9ce @m f'@r1 /.k f@r1_1 ,u1(f',
.(f@r1.)) .% ,u @l is empty4 ,let @m @;,b' @l 3si/ ( all s+letons on ;,j
tgr ) ^? memb]s @m ,u;n"(f, t) @l ( @m @;,b @l ": @m .(f@r1.) @e t_4 @l
,!n @m @;,b' @l is a clop5 base = ;,y4 $p ,6see t ;,y is normal1 let ;,a &
;,b 2 4jo9t clos$ subsets ( ;,y4 ,z 9 ! analog\s pro( = @m ,y(4)_1 @l we
assume )\t loss ( g5]al;y t bo? ;,a & ;,b >e 3ta9$ 9 ;,f4 ,= ea* ;f 9 ;,a1
let @m c(f) @l 2 ! smalle/ 9teg] ;n s* t @m ,u;n"(f, t) @l avoids ;,b = "s
;t 9 @m ,t;n_1 @l & let @m t(f) @l 2 s* a ;t4 ,simil>ly1 = ea* ;f 9 ;,b1
let @m c(f) @l 2 ! smalle/ 9teg] ;n s* t @m ,u;n"(f, t) @l avoids ;,a = "s
;t 9 @m ,t;n_1 @l & let @m t(f) @l 2 s* a ;t4 ,= ea* @m n .1: #1_1 let
,a;n .k .(f@rn\ f @e ,a @l & @m c(f) .k n.) @l & @m ,b;n .k .(f@rn\ f @e
,b @l & @m c(f) .k n.)_4 @l ,= ea* ;f 9 @m ,a .+ ,b @l ) @m c(f) .k n_1
let t'(f) .k t(f) .+ .(,a;i"\ #1 "k: i "k: n.) .+ .(,b;i"\ #1 "k: i "k:
n.)_4 @l ,n[ let @m ,u .k .+ .(,u;c(f)"(f, t'(f))\ f @e ,a.) @l & @m ,v .k
.+ .(,u;c(f)"(f, t'(f))\ f @e ,b.)_4 @l ,cle>ly ;,u & ;,v >e op5 sets :
3ta9 ;,a & ;,b1 respectively4 $p ,c]ta9ly no po9t ( ;,f c 2 9 @m ,u .%
,v_4 @l ,6see t ;,u & ;,v >e 4jo9t1 suppose t @m y .k (.(.s, .t.), r) @e
,u .% ,v_4 @l ,!n @m y @e ,u;c(f)"(f, t'(f)) .% ,u;c(f')"(f', t'(f')) @l =
"s ;f 9 ;,a & @m f' @l 9 ;,b4 ,)\t loss ( g5]al;y assume t @m f@rc(f) .k
.s@rc(f), f'@rc(f') .k .t@rc(f'), @l & @m c(f) "k: c(f')_4 ,let n .k
c(f)_4 @l ,!n @m ,a;n @l 2l;gs 6bo? @m t'(f) @l & @m t'(f')_4 @l ,s @m
,a;n @e r @l s9ce @m f@rn @e ,a;n_1 @l b @m ,a;n /@e r @l s9ce @m f'@rn
/@e ,a;n_4 @l ,s9ce ? is impossi#1 ;,u & ;,v >e 4jo9t4 ,"!=e ;,y is ,t4
 
@l $p 7b73 ;,y has "* @m 2^.w^;1 @l at ea* po9t ( ;,f4 ,al? ;,y "!=e is n
a ,moore space & does n h a @m .s-_ @l 4jo9t or a @m .s-_ @l loc,y c.ta#
base1 we c reflect ! c.ta# /ructure t ;,y does h4 ,giv5 any @m n .1: #1_1
let @;,g;n^1 .k .(,u;n"(f, .f)\ f @e ,f.), @;,g;n^2 .k .(.(y.)\ y @e ,j-
.+ @;,g;n^1, @l & @m @;,g;n .k @;,g;n^1 .+ @;,g;n^2_4 @l ,once ag1* is
loc,y c.ta#1 z @m @;,g;n+1 @l te/ifies4 ,h["e1 z = ;,x1 @m " .+%n_ .1:_
#1] @;,g;n @l is n a base = ;,y4 $p 7c73 ,we n[ modify ! pro( t ;,x is
p>a-,l9del( 6prove t ;,y is p>a-,l9del(1 z 9 ,*apt] #d4 ,giv5 an op5 cov]
@m @;,v @l ( @m ;,y1 @l = ea* ;f 9 ;,f let @m c(f) @l 2 ! smalle/ 9teg] ;n
s* t @m ,u;n"(f, t) @l is 3ta9$ 9 a memb] ( @m @;,v @l = "s ;t 9 @m ,t;n_4
@l ,once ag1 = any @m n .1: #1_1 let ,a;n .k .(f@rn\ f @e ,f @l & @m c(f)
.k n.); @l = ea* @m .r @l 9 @m ,a;n_1 @l *oose @m t;.r @l 9 @m ,t;n @l s t
@m ,u;n"(f, t;.r") @l ref9es a memb] ( @m @;,v @l = any ;f 9 @m @(.r_3
,f@)_4 @l ,= 3v5i;e1 = ea* ;f 9 ;,f1 let @m t(f) .k t;.r @l ": @m .r .k
f@rc(f); let t'(f) .k t(f) .+ .(,a;i"\ #1 "k: i "k: c(f).)_4 @l $p ,n[1 =
ea* ;f 9 @m ;,f1 let @m ,w(f) @l 2 ! op5 set @m ,u;c(f)"(f, t'(f))_1 @l :
is det]m9$ by @m f@rc(f)_4 @l ,def9e @m @;,w^1, @;,w^2, @l &! op5 ref9e;t
@m @;,w @l = @m @;,v @l z 9 ! analog\s pro( = @m ,y(4)_4 @l ,z 9 ! pro( =
;,x1 @m ,w(f) @l & @m ,w(f') @l >e 4jo9t if @m c(f) /.k c(f') @l ": ;f &
@m f' @l >e 9 ;,f4 ,z a result1 @m @;,w @l is loc,y c.ta#2 = ea* ;f 9 ;,f1
! op5 set @m ,z(f) .k ,w(f) .% ,u;c(f)+1"(f, .f) @l wit;ses ! local
c.tabil;y ( @m @;,w @l at ;f4 $p 7d7 - 7f73 ,! pro( t ;,x is n normal c 2
modifi$ 6prove t ;,y is n collec;nwise normal1 j z 9 ,*apt] #d2 ? is left
6! r1d]4 ,simil>ly1 ! pro(s %[+ t ;,x is metacompact & c.tably p>acompact
c 2 modifi$ 6prove t ;,y is metacompact & c.tably p>acompact1 z 9 ,*apt]
#d4 @m $4