@l $f $$c ,sec;n #b.a4 ,3v5;ns4 $p ,except ": o!rwise specifi$1 ! lrs ;i,
;j, ;m, & ;n w 2 res]v$ = nonnegative 9teg]s4 ,! symbol @m _"k @l w denote
9clu.n1 n necess>ily prop] 9clu.n4 ,=! m1n+ 82l;g+ to10 ! symbol @m @e @l
&! ^w 890 w 2 us$ 9t]*angeably4 ,! symbol @m @r @l w denote re/ric;n4
,ord]$ pairs w 2 notat$ ) ord9>y p>5!ses4 $p ,! symbol @m _< @l w 2 us$ )!
m1n+ 8ti$ to0 z foll[s4 ,let ;g 2 a func;n ) doma9 an ord9al .g &) range a
set ( ord9als4 ,= an ord9al @m .a_1 g_<.a @l w denote ! new func;n @m g'
.k g .+ .((.g, .a).)_4 @l ,t is1 @m g' @l has doma9 @m .g+1, g'\.g .k g,
@l & @m g'(.g) .k .a_4 @l $p ,! symbol @m $4 @l w m>k ! 5d ( ea* def9i;n1
example1 or pro(4 ,possibly unfamili> topological prop]ties appe>+ 9 !
text w 2 def9$ 9 ! gloss>y2 "s (! less famili> "os w 2 def9$ 9 ! text z
well4 $p ,! set !ory assum$ 9 ? pap] is ,,zfc4 ,results requir+ a4i;nal
axioms ( set !ory w 2 s 9diicat$4 ,9 "picul>1 "s (! examples use @m
,,ma(.w1)_1 @l "p ( @m (,,ma+@w,,ch) @l 7": ,,ma /&s = ,m>t9's axiom &
,,ch =! 3t9uum hypo!sis74 ,a topological equival5t ( @m ,,ma(.w1) @l is3
,9 a -pact1 c4c4c4 ,hausdorff space1 ! 9t]sec;n ( any collec;n ( @m
.w1-many @l d5se op5 sets is nonempty4 ,note t a topological space is 7has
!7 c4c4c4 7c.ta# *a9 3di;n7 provid$ "! is no collec;n ( unc.tably _m
mutu,y 4jo9t op5 subsets (! space4
 
@l $p $$c ,sec;n #b.b ,p>a-,l9del( ,spaces3 ,basic ,facts $p $$ub #b.b.a3
,def9i;ns4 $$uf $p 7a7 ,a topological space ;,x is $$ub p>a-,l9del( $$uf
if e op5 cov] ( ;,x has a loc,y c.ta# op5 ref9e;t4 ;,x is @m
".s-para-,linelof%:] @l if e op5 cov] ( ;,x has an op5 ref9e;t : is @m
.s-_ @l loc,y c.ta# 7i4e41 ! union ( c.tably _m loc,y c.ta# collec;ns74 $p
7b7 ,a collec;n @m @;,d @l ( subsets (a topological space ;,x is $$ub
4crete $$uf if e po9t ( ;,x has a nei<borhood : meets 7i4e41 has nonempty
9t]sec;n )7 at mo/ "o memb] ( @m @;,d_4 @l ,a collec;n ( po9ts ( ;,x is
$$ub 4crete $$uf if ! associat$ collec;n ( s+letons is 4crete4 $p 7c7 ,a
@m ,t1-space @l ;,x @l is $$ub collec;nwise ,hausdorff $$uf 7respectively1
$$ub /r;gly collec;nwise ,hausdorff7 $$uf if = e 4crete collec;n @m
.(x;.a"\ .a @e ,a.) @l ( po9ts ( ;,x "! is a mutu,y 4jo9t 7respectively1
4crete7 collec;n @m .(,u;.a"\ .a @e ,a.) @l ( op5 sets ) @m x;.a @e ,u;.a
@l = ea* @m .a @l 9 @m ;,a_4 $4 @l $p ,! many sep>,n prop]ties :
p>acompact;s implies = ,hausdorff spaces provide a gd m1ns = -p>+
p>acompact & p>a-,l9del( @m ,t3-spaces_4 @l ,a m -plete tr1t;t (! positive
sep>,n results = p>a-,l9del( @m ,t3-spaces @l is giv5 0,fleissn] & ,re$ 9
@m @(2@)_4 @l ,"o ( _! important results ( ? k9d is t ^! spaces >e /r;gly
collec;nwise ,hausdorff4 ,a pro( ( ?1 : 9dicates ! flavor ( >gu;ts us+
local c.tabil;y1 is 9clud$ "h4 $p $$ub #b.b.b3 ,lemma4 $$uf ,let ;,h & ;,k
2 subsets (a topological space @m ;,x_4 @l ,= any ;h 9 ;,h & ;k 9 ;,k1 say
@m h @# k @l & @m k @# h @l provid$ @m h /.k k_4 @l ,= ea* ;x 9 @m ,h .+
,k_1 let ,s(x) @l 2 an op5 nei<borhood ( ;x4 ,suppose t = ea* ;x 9 @m ,h
.+ ,k_1 @l "! is no ele;t @m x' @l ( @m ,h .+ ,k @l s* t @m x' @# x & x'
@e ",s(x)<:]_4 @l ,suppose1 al1 t = ea* ;x 9 @m ,h .+ ,k_1 @l "! >e only
c.tably _m po9ts @m x' @l 9 @m ,h .+ ,k @l = : @m x @# x' @l & @m ,s(x) .%
,s(x') /.k .f_4 @l ,!n ea* @m ,s(x) @l c 2 ref9$ 6an op5 nei<borhood @m
,r(x) @l ( ;x s t ! collec;n @m .(,r(x)\ x @e ,h .+ ,k.) @l satisfies !
foll[+4 ,= ea* set @m ,r(x)_1 @l "! is no set @m ,r(x') @l s* t x @m @# x'
@l & @m ,r(x) .% ,r(x') /.k .f_4 @l $p $$ub ,pro(4 $$uf ,let @m @: @l 2 !
equival;e rel,n on @m ,h .+ ,k @l g5]at$ 0! rule @m x @: x' if x @# x' @l
& @m ,s(x) .% ,s(x') /.k .f_4 ,let ,e .k (,h .+ ,k)_/@:_1 @l ! set ( @m
@:-_ @l equival;e classes on @m ,h .+ ,k_4 @l ,notice t ea* equival;e
class is c.ta#4 ,5um]ate ! memb]s ( ea* s* class ;e z @m x;e[0, x;e[1,
x;e[2, ''' @l 7f9itely many or @m .w-many @l z ne$$74 ,= ea* @m x;e[n_1
let ,r(x;e[n") .k ,s(x;e[n") - .+ .(",s(x;e[j")<:]\ j "k n @l & @m x;e[j
@# x;e[n".)_4 @m $4
 
@l $p $$ub #b.b.c3 ,!orem4 $$uf ,let ;,x 2 a p>a-,l9del( @m ,t3-space_4 @l
,!n3 $p 7a7 ;,x is collec;nwise ,hausdorff4 $p 7b7 ;,x is /r;gly
collec;nwise ,hausdorff4 $p $$ub ,pro(4 $$uf 7a73 ,let @m ,x0 .k .(x;.a"\
.a @e ,a.) @l 2 a 4crete collec;n ( po9ts ( ;,x4 ,0regul>;y ( ;,x1 = ea*
@m .a @l 9 ;,a let @m ,u;.a @l 2 an op5 nei<borhood ( @m x;.a @l s* t @m
",u;.a"<:] .% ,x0 .k .(x;.a".)_4 @l ,s9ce ;,x is p>a-,l9del(1 ! op5 cov]
@m .(,u;.a"\ .a @e ,a.) .+ .(,x-,x0.) @l has a loc,y c.ta# op5 ref9e;t @m
@;,v_4 @l ,= ea* @m .a @l 9 ;,a let @m ,v;.a @l 2 a nei<borhood ( @m x;.a
@l 2l;g+ to @m @;,v_4 @l ,s9ce ! collec;n @m @;,v0 .k .(,v;.a"\ .a @e ,a.)
@l is loc,y c.ta#1 = ea* @m .a @l 9 ;,a let @m ,v';.a @l 2 an op5 subset (
@m ,v;.a @l : wit;ses ? at @m x;.a; @l t is1 @m x;.a @e ,v';.a @l & @m
,v';.a @l meets only c.tably _m memb]s ( @m @;,v0_4 @l ,!n ! collec;n (
op5 sets @m @;,v'0 .k .(,v';.a"\ .a @e ,a.) @l is />-c.ta#2 t is1 ea* set
@m ,v';.a @l avoids all b c.tably _m memb]s ( @m @;,v';0_4 @l $p ,n[ we c
apply ,lemma #b.b.b ) @m ,h .k ,x0, ,k .k ,x0, @l & @m ,s(x;.a") .k ,v';.a
@l = ea* @m .a @l 9 @m ;,a; @l let ! sets @m ,r(x;.a") _"k ,s(x;.a") @l
satisfy ! lemma's 3clu.n4 ,let @m ,w;.a .k ,r(x;.a") @l = ea* @m .a @l 9
;,a4 ,!n @m @;,w0 .k .(,w;.a"\ .a @e ,a.) @l is a collec;n ( mutu,y 4jo9t
op5 sets ) @m x;.a @e ,w;.a @l = ea* @m .a @l 9 ;,a4 ,"!=e ;,x is
collec;nwise ,hausdorff4 $p 7b73 ,3t9ue f 7a74 ,t2 an op5 cov] : te/ifies
t @m @;,w0 @l is loc,y c.ta# 7= example1 any op5 cov] wit;s+ ! local
c.tabil;y ( @m @;,v0)_4 @l ,n[ let @m @;,z @l 2 a loc,y c.ta# op5 ref9e;t
( @m @;,w'_4 @l ,= ea* .a 9 ;,a let @m ,z';.a @l 2 an op5 set wit;s+ !
local c.tabil;y ( @m @;,z at x;.a_2 @l us+ ! regul>;y ( ;,x1 assume )\t
loss ( g5]al;y t @m ",z';.a"<:] _"k ,w;.a_4 @l $p ,6prep>e = an applic,n (
,lemma #b.b.b1 let @m ,h .k ,x0 @l & @m ,k .k ,x- .+ @;,w0_4 @l ,= ea* @m
.a @l 9 ;,a let @m ,s(x;.a") .k ,z';.a_4 @l ,f9,y1 = ea* ;x 9 ;,k let @m
,s(x) @l 2 a memb] ( @m @;,z @l ) @m x @e ,s(x)_4 @l ,s9ce ^! satisfy !
hypo!ses (! lemma1 let ! sets @m ,r(x) @l = ;x 9 @m ,h .+ ,k @l 2 z 9 xs
3clu.n4 ,n[ let @m ,g;.a .k ,r(x;.a") @l = ea* @m .a @l 9 ;,a & @m @;,g0
.k .(,g;.a"\ .a @e ,a.)_4 @l ,if @m x @e ,k_1 @l !n @m ,r(x) @l meets no
memb] ( @m @;,g0_4 @l ,o!rwise1 = "o @m .a @l 9 ;,a1 x lies 9 @m ,w;.a_1
@l : meets j "o memb] ( @m @;,g0--_ @l "nly @m ,g;.a_4 @l ,?us @m @;,g0 @l
is 4crete4 ,"!=e ;,x is /r;gly collec;nwise ,hausdorff4 @m $4 @l $p ,a
numb] ( results ab p>acompact;s su7e/ simil> 3sid],n ( p>a-,l9del(;s4 ,=
example1 0a famili> !orem ( ;,e4 ,
 
@l $p $$ub #b.b.d3 ,example4 $$uf ,! ,moore space 3/ruct$ 0,william
,fleissn] 9 ,7#a7' is @m .s-_ @l p>a-,l9del( b n p>a-,l9del(4 @m $4 @l $p
$$ub #b.b.e3 $$uf ,!orem 7,fleissn] & ,re$ ,7#b7'74 ,if a @m ,t3-space @l
is c.tably p>acompact & @m .s-_ @l p>a-,l9del(1 !n x is p>a-,l9del(4 @m $4
@l $p ,"s /&>d !orems ab pres]v,n ( p>acompact;s translate 6simil> !orems
ab pres]v,n ( p>a-,l9del(;s4 $p $$ub #b.b.f3 ,!orems4 $$uf 7a7 ,a clos$
subspace (a p>a-,l9del( space is p>a-,l9del(4 ,if ! space is @m ,t3 @l &
c.tably p>acompact1 we c replace clos$ by @m ,f;.s @l 7us+ ! ju/-m5;n$
result ( ,fleissn] & ,re$74 $p 7b7 ,! product (a p>a-,l9del( space )a
-pact space is ag p>a-,l9del(4 @m $4 @l $p ,likewise1 "s /&>d examples (
nonpres]v,n ( p>acompact;s al rev1l nonpres]v,n ( p>a-,l9del(;s4 $p $$ub
#b.b.g3 ,examples4 $$uf 7a7 ,an op5 subspace (a -pact1 ,hausdorff space
ne$ n 2 p>a-,l9del(4 ,an example ( ? is @m .w1 @l z a subspace (! ord9al
space @m .w1+1_4 @l ,9 fact1 @m .w1 @l is n ev5 meta-l9del(4 $p 7b7 ,!
product (a ,l9del( space )a ,l9del( metric space ne$ n 2 p>a-,l9del(4 ,an
example ( ? is ,mi*ael's ,product ,topology ,7#aa1 ;p4 #aje7'4 @m $4 @l $p
,ano!r prop]ty us+ local c.tabil;y1 t ( hav+ a @m .s-_ @l loc,y c.ta#
base1 is closely l9k$ ) metrizabil;y4 ;,v4 ,f$orcuk ,7#aj7' prov$ t e
p>acompact @m ,t2-space @l hav+ a @m .s-_ @l loc,y c.ta# base is actu,y
metriza#4 ,l ! prop]ty ( 2+ p>a-,l9del(1 ? prop]ty is n "u/oood z well z
xs local f9ite;s c.t]"p 7i4e41 hav+ a @m .s-_ @l loc,y f9ite base1 : = @m
,t3-spaces @l is equival5t 62+ metriza#74 ,f9,y1 nei ( ^! two local
c.tabil;y prop]ties implies ! o!r4 $p $$ub #b.b.h3 ,examples4 $$uf 7a7 ,!
,moore space 3/ruct$ 0,fleissn] 9 ,7#a7' has a @m .s-_ @l loc,y c.ta# base
b is n p>a-,l9del(4 $p 7b7 ,0,f$orcuk's !orem1 any p>acompact b
nonmetriza# @m ,t2-space @l is p>a-,l9del( b does n h a @m .s-_ @l loc,y
c.ta# base4 ,an example ( ? is ! ,sorg5frey l9e4 @m $4 @l $p ,h["e1 z
,fleissn] & ,re$ po9t \ 9 ,7#b7'1 x is easy 6see ! foll[+4 $p $$ub #b.b.i3
,!orem4 $$uf ,e p>a-,l9del( ,moore space has a @m .s-_ @l loc,y c.ta#
base4 @m $4 @l