Publicacions Ma em`a iques, Vol 39 (1995), 215–232.
THE FREUDENTHAL SPACE
FOR APPROXIMATE SYSTEMS
OF COMPACTA AND SOME APPLICATIONS
I an Lonˇ
ca
Abs ac
In his pape we define a space σ(X) o app oxima e sys ems
o compac spaces. The cons uc ion is due o H. F euden hal
o usual in e se sequences [4, p. 153–156]. We s ablish he ol-
lowing p ope ies o his space: (1) The space σ(X) is a pa a-
compac space, (2) Mo eo e , i Xis an app oxima e sequence
o compac (me ic) spaces, hen σ(X) is a compac (me ic)
space (Lemma 2.4). We gi e he ollowing applica ions o he
space σ(X): (3) I Xis an app oxima e sys em o con inua, hen
X= lim Xis a con inuum (Theo em 3.1), (4) I Xis an app ox-
ima e sys em o he edi a ily unicohe en spaces, hen X= lim X
is he edi a ily unicohe en (Theo em 3.6), (5) I Xis an app oxi-
ma e sys em o ees wi h mono one on o bonding mappings, hen
X= lim Xis a ee (Theo em 3.13).
1. In oduc ion
Le Ube any co e ing o a space X. Fo any subse Yo Xwe define
S (Y,U)=∪{U∈U:U∩Y=∅}.
Simila ly, we define S U={S (U, U):U∈U}. Induc i ely, o each
posi i e in ege n,S
nU= S (S n−1U), whe e S 1U=S U.
We say ha a co e Vis a s a efinemen o aco e Ui he co e
S Vis a efinemen o U.
An open co e Wo a space Xis no mal [3, p. 379] i he e exis s a
sequence W1,W2,... o open co e s o he space Xsuch ha W1=W
and Wi+1 is a s a efinemen o Wi o i=1,2,... AT1space Xis
pa acompac iff each open co e o Xis no mal [3, Theo em 5.1.12]. A
T1space Xis no mal iff each locally fini e open co e o Xis no mal [3,
p. 379].
The se o all no mal co e s o Xis deno ed by Co (X).
216 I. Lonˇ
ca
I U,V∈Co (X) and V efines U, we w i e V≺U.I ,g:Y→X
a e U-nea mappings, i.e. i o any y∈Y he e exis s U∈Uwi h (y),
g(y)∈U, we w i e ( ,g)≺U.
App oxima e in e se sys ems we e in oduced by S. Ma deˇsi´c and
L. R. Rubin [11] o compac a and by S. Ma deˇsi´c and Wa anabe [12]
o gene al opological spaces.
Defini ion 1.1. An app oxima e in e se sys em X={Xa,Ua,p
ab,A}
consis s o he ollowing da a: A p eo de ed se (A, ≤) which is di ec ed
and has no maximal elemen ; o each a∈A, a opological space Xaand
a no mal co e ing Uao Xa(called he mesh o Xa) and o each pai
a≤b om A, a mapping pab :Xb→Xa. Mo eo e he ollowing h ee
condi ions mus be sa isfied:
(A1) The mappings pabpbc and pac a e Ua-nea , a≤b≤c, i.e.
(pabpbc,p
ac)≺U
a.
(A2) Fo each a∈Aand each no mal co e U∈Co (Xa) he e is b≥a
such ha (pacpcd,p
ad)≺U, whene e a≤b≤c≤d.
(A3) Fo each a∈Aand each no mal co e U∈Co (Xa) he e is b≥a
such Uc≺p−1
ac (U)={p−1
ac (U):U∈U} o each c≥b.
In he case o me ic compac spaces we eplace he no mal co e ings
by eal numbe s [11].
I he spaces Xaa e T1pa acompac , hen in he abo e defini ion one
can use all open co e ings on he spaces Xa,a∈A, since in his case
each open co e is no mal.
Defini ion 1.2. An app oxima e map p={pa:a∈A}:X→Xa
in o an app oxima e in e se sys em X={Xa,Ua,p
ab,A}is a collec ion
o maps pa:X→Xa,a∈A, such ha he ollowing condi ion holds
(AS) Fo any a∈Aand any U∈Co (Xa) he e is b≥asuch ha
(pacpc,p
a)≺U o each c≥b. (See [12].)
Defini ion 1.3. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em and le p={pa:a∈A}:X→Xabe an app oxima e map. We
say ha pis a limi o Xp o ided i has he ollowing uni e sal p ope y
[12, p. 592]:
(UL) Fo any app oxima e map q={qa:a∈A}:Y→Xao a space
Y he e exis s a unique map g:Y→Xsuch ha pag=qa o
any a∈A.
Rema k 1.4. I p:X→Xis a limi o X, hen he space Xis
de e mined up o a unique homeomo phism. The e o e, we o en speak
o he limi Xo Xand we w i e X= lim X.
The F euden hal space o app oxima e sys ems 217
Defini ion 1.5. Le X={Xa,Ua,p
ab,A}be an app oxima e sys em.
A poin x=(xa)∈{Xa:a∈A}is called a h ead o Xp o ided i
sa isfies he ollowing condi ion:
(L) (∀a∈A)(∀U ∈Co (Xa))(∃b≥a)(∀c≥b)pac(xc)∈s (xa,U).
Rema k 1.6. I Xais a T3.5space, hen he se s s (xa,U), U∈
Co (Xa), o m a basis o he opology a he poin xa. The e o e, o
an app oxima e sys em o Tychonoff spaces condi ion (L) is equi alen
o he ollowing condi ion:
(L)∗(∀a∈A) lim{pac(xc):c≥a}=xa.
The ollowing heo em shows ha he se o h eads is a limi o X.
Theo em 1.7. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em. Le X⊂Xabe he se o all h eads o Xand le pa:X→Xa
be he es ic ion pa=πa|Xo he p ojec ion πa:Xa→Xa,a∈A.
Then p={pa:a∈A}X→Xis a limi o X.
P oo : See [12, Theo em (1.14)].
The canonical limi o Xis he se o all h eads o X[12, p. 593].
Theo em 1.8. Fo any app oxima e in e se sys em X he canonical
limi lim Xis closed in Xa. Mo eo e , i all Xaa e compac and
non-emp y, hen lim Xis compac and non-emp y.
P oo : See he p oo o Lemma (1.16) and Theo em (4.1) o [12].
Lemma 1.9. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o Tychonoff spaces, le Xbe he canonical limi o Xand le
B⊆Abe a cofinal subse o A. Then he collec ion Bo all se s o
he o m p−1
b(Ub), whe e b∈Band Vb⊆Xbis open, is a basis o he
opology o X.
P oo : See [12, (1.18) Lemma].
Theo em 1.10. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o compac Hausdo ff spaces wi h limi X. Fo each closed F⊆X
we ha e
F={p−1
a(pa(F)) : a∈A}.
P oo : I is ob ious ha F⊆p−1
a(pa(F)) o each a∈A. Thus, F⊆
∩{p−1
a(pa(F)) : a∈A}.I x/∈F, hen, by Lemma 19 we in e ha he e
exis s an a∈Aand an open se Ua⊆Xasuch ha x∈p−1
a(Va)⊆X−F.
This means ha pa(x)/∈pa(F) and x/∈p−1
a(pa(F)).
218 I. Lonˇ
ca
2. The F euden hal space σ(X)
The ollowing cons uc ion is simila o he cons uc ion due o
H. F euden hal [4, p. 153] o usual in e se sequences. Fo any usual
in e se sys em see [10].
Le X={Xa,Ua,p
ab,A}be an app oxima e in e se sys em o compac
Hausdo ff spaces wi h limi Xand he p ojec ions pa:X= lim X→Xa.
The F euden hal space σ(X) associa ed o Xis he se
(1) σ(X)=X{Xa:a∈A}
whe e all Xaand hei limi Xa e conside ed as being disjoin se s [10],
in which a opology is defined as ollows. I Uais an open se in Xa, le
(2) U∗
a={p−1
ab (Ua):b≥a}p−1
a(Ua).
Now, we define a opology Ton σ(X) by a base [3, p. 27] Bwhich consis s
o all open se s Uain all Xaand all U∗
a o all open se s Ua⊆Xa,a∈A.
Since he se s p−1
a(Ua) o m a basis o X, i ollows ha Bisaco e o
σ(X). By i ue o [3, p. 27] we need o p o e ha o each x∈σ(X)
and each pai B,C∈Bwi h x∈B∩C he e is a D∈Bsuch ha
x∈D⊆B∩C. I suffices o p o e his s a emen i Bis some U∗
a
and Cis some U∗
b.I xis a poin o Xc, hen xis con ained in a se
p−1
ac (Ua)∩p−1
bc (Ub) which is open in Xcand hus belongs o B.I xis a
poin o X, hen
(3) z∈p−1
a(Ua)p−1
b(Ub)
i.e., xa=pa(x)∈Ua, and xb=pb(x)∈Ub. Choose Va∈Co (Xa),
Vb∈Co (Xb) such ha
(4) S (xa,Va)⊆Uaand S (xb,Vb)⊆Ub.
Take Wa∈Co (Xa), Wb∈Co (Xb) such ha S 2Wa≺V
a,S
2Wb≺
Vband c∈Asuch ha c≥a,b, (A2) and (A3) hold o a,b,Wa,Wb
and (L) holds o x,a,b,Wa,Wb. Pu
(5) Vc= S (xc,Uc).
Since x∈p−1
c(Vc)⊆V∗
c, he p oo will be comple e i we show ha
(5.1) V∗
c⊆U∗
aU∗
b.
The F euden hal space o app oxima e sys ems 219
We fi s p o e ha
(6) p−1
c(Vc)⊆p−1
a(Ua)p−1
b(Ub).
Conside a poin y=(ya)∈p−1
c(Vc). By (5) he e is a U1∈U
csuch
ha
(7) xc,y
c∈U1.
By he choice o c(p ope y (A3)) Uc≺p−1
ac (Wa) and Uc≺p−1
bc (Wb).
This means ha he e is a W1∈W
aand W2∈W
bsuch ha U1⊆
p−1
ac (W1) and U1⊆p−1
bc (W2). Thus, (7) implies
(8) pac(xc),p
ac(yc)∈W1and pbc(xc),p
bc(yc)∈W2.
By he choice o c(p ope y (L)), he e a e W3∈W
a,W4∈W
bsuch
ha
(9) xa,p
ac(xc)∈W3and xb,p
bc(xc)∈W4.
Since y∈p−1
b(Ub)⊆X, he e is a d≥csa is ying (L) o y,a,Waand
o y,b,Wb. Thus, he e exis a W5∈W
a,W6∈W
band U4∈U
csuch
ha
(10) pad(yd),y
a∈W5and pbd(yd),y
b∈W6
and
(11) pcd(yd),y
c∈U4.
By he choice o c(p ope y (A3)), Uc≺p−1
ac (Wa) and Uc≺p−1
bc (Wb).
Hence, he e exis a W7∈W
aand W8∈W
bsuch ha U4⊆p−1
ac (W7)
and U4⊆p−1
bc (W8). By (11) we ha e
(12) pacpcd(yd),p
ac(yc)∈W7and pbcpcd(yd),p
bc(yc)∈W8.
By he choice o c(p ope y (A2)), we also ha e a W9∈W
aand W10 ∈
Wbsuch ha
(13) pacpcd(yd),p
ad(yd)∈W9and pbcpcd(yd),p
bd(yd)∈W10.
Now, (9), (8), (12), (13), (10), S 2Wa≺V
aand S 2Wa≺V
byield a
V∈V
aand a V ∈V
bsuch ha xa,ya∈W1∪W3∪W5∪W7∪W9⊆V
and xb,yb∈W2∪W4∪W6∪W8∪W10 ⊆V. This and (4) imply

220 I. Lonˇ
ca
pa(y)=ya∈S (xa,Va)⊆Uaand pb(y)=yb∈S (xb,Vb)⊆Ub. This
means ha y∈p−1
a(Ua)∩p−1
b(Ub), i.e., (6) is p o ed. I emains o p o e
(14) p−1
cd (Vc)⊆p−1
ad (Ua)p−1
bd (Ub)∀d≥c.
Le zd∈p−1
cd (Vc). By (5) he e is a U11 ∈U
csuch ha
(15) xc,p
cd(zd)∈U11.
By he choice o c(p ope y (A3)) he e is a W11 ∈W
aand a W12 ∈W
b
such ha U11 ⊆p−1
ac (W11) and U11 ⊆p−1
bc (W12). Thus, (15) implies
(16) pac(xc),p
ac(pcd(zd)) ∈W11 and pbc(xc),p
bc(pcd(zd)) ∈W12.
By (A2) we in e he e a e W13 ∈W
aand W14 ∈W
bsuch ha
(17) pacpcd(zd),p
ad(zd)∈W13 and pbcpcd(zd),p
bd(zd)∈W14.
F om (9), (16) and (17) i ollows xa,pad(zd)∈S Vaand xb,pbd(zd)∈
S Vb. By (4) pad(zd)∈Uaand pbd(zd)∈Ub. We in e ha zd∈
p−1
ad (Ua)∩p−1
bd (Ub) and (14) is p o ed. Hence, we ha e x∈V∗
c⊆U∗
a∩U∗
b,
i.e., (5.1) is p o ed. This means ha Bis a basis o some opology T
on σ(X).
Now, we will p o e ha Tis a Hausdo ff opology. Le x,ybe a pai
o dis inc poin s in σ(X). I x,y/∈lim X, hen he e exis s a pai a,
b∈Asuch ha x∈Xa,y∈Xb.I a=b, hen xand yha e disjoin
neighbo hoods since Xais a Hausdo ff space. I a=b, hen Xaand
Xba e disjoin neighbo hoods (in σ(X)) o xand y espec i ely. Now,
suppose ha x∈lim Xand y/∈lim X. Le y∈Xb o some b∈A.
By i ue o Lemma 1.9 he e is a c>band an open se Ucsuch ha
p−1
c(Uc) is a neighbo hood o xin lim X. I is clea ha Xband V∗
ca e
disjoin neighbo hoods o yand xin σ(X). Finally, le x,y∈lim X.
Since lim Xis a Hausdo ff space, he e a e open (in lim X) disjoin se s
Uand Vsuch ha x∈Uand y∈V. By i ue o Lemma 1.9 he e
exis s a b∈Aand open se s Uband Vbsuch ha x∈p−1
b(Ub)⊆Uand
y∈p−1
b(Vb)⊆V. I ollows ha Uband Vba e disjoin since Uand V
a e disjoin . Hence, U∗
band V∗
ba e disjoin . Thus, σ(X) is a Hausdo ff
space.
Ane in a opological space X[3, p. 73] is an a bi a y unc ion om
a non-emp y di ec ed se D o he space X. Ne s will be deno ed by
N={xd:d∈D}. A poin x∈Xis called a limi o a ne N={xd:
d∈D}i o e e y neighbo hood Uo x he e is a d0∈Dsuch ha
xd∈U o each d≥d0. We say ha he ne Ncon e ges o x. A poin
x∈Xis called a clus e poin o a ne N={xd:d∈D}i o e e y
neighbo hood Uo xand e e y d0∈D he e exis s a d≥d0such ha
xd∈U.
The F euden hal space o app oxima e sys ems 221
Lemma 2.1. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o non-emp y compac Hausdo ff spaces wi h limi X.
1. I Ais a cofinal subse o A, hen each amily N={xa:xa∈
Xa,a∈A}is a ne in σ(X)which has a leas one clus e poin
x(in he opology T) such ha x∈X⊆σ(X).
2. Each poin x∈Xis he limi (in he opology T) o he ne
{pa(xa):a∈A}.
P oo : Fo each a∈Awe conside he ne Na={pab(xb):b∈A,b≥
a}. F om he compac ness o Xai ollows ha he se Cao all clus e
poin s o Nais non-emp y. Clea ly, each Cais closed and compac in
Xa. Fi s , we p o e
(a) Fo each a∈AC
ais a non-emp y subse o pa(X).
I we suppose ha some ca∈Ca pa(X), hen caand pa(X) espec i ely,
ha e disjoin neighbo hoods Uand V. By i ue o he p ope y (B3)
[12, p. 606, 615] he e is a b≥asuch ha pac(Xc)⊆V o each c≥b,
c∈A. This is impossible since he e exis s c≥bsuch ha pac(xc)∈U
(cais a clus e poin o he ne Na).
F om (a) i easily ollows ha
(b) Fo each a∈A he se p−1
a(Ca)is non-emp y.
By (b) he e is ya∈p−1
a(Ca)⊆lim X,a∈A. Since lim Xis compac ,
he e is a clus e poin y∈lim Xo he ne Y={ya:a∈A}. Le us
p o e
(c) pa(y)∈Ca,a∈A.
I suffices o p o e ha o each neighbo hood Uao pa(y) and each
b0 he e exis s a d≥b0such ha pad(xd)∈Ua. Le Ube a no mal co e
o Xasuch ha
(18) S 2(pa(y),U)⊆Ua.
Le U1∈Ube such ha pa(y)∈U1. Then p−1
a(U1) is a neighbo hood
o y. The se Bo all b∈Awi h yb∈p−1
a(U1) is cofinal in Asince y
is a clus e poin o Y. By i ue o (AS) he se B⊆Bo all b∈B,
b≥b0, such ha
(19) (pa,p
abpb)≺U
is cofinal in A. Simila ly, by (A2), he se B ⊆Bo all b∈Bsuch
ha
(20) (pac,p
abpbc)≺U,c≥b
222 I. Lonˇ
ca
is cofinal in A. Le b∈B. Then yb∈p−1
a(U1). Thus
(21) pa(y),p
a(yb)∈U1.
By i ue o (19) i ollows
(22) pa(yb),p
abpb(yb)∈U2∈U.
This and (21) imply
(23) pabpb(yb)∈S (pa(y),U).
Now, pb(yb)∈Cbsince yb∈p−1
b(Cb). We in e ha p−1
ab (S (pa(y),U)) is
a neighbo hood o pb(yb). Since pb(yb) is a clus e poin o Na he e is
ad≥b≥b0,d∈Asuch ha pbd(xd)∈p−1
ab (S (pa(y),U)). This means
ha pab(pbd(xd)) ∈S (pa(y),U). Using (20), pad(xd)∈S 2(pa(y),U).
Thus, by (18)
(24) pad(xd)∈Ua.
We in e ha pa(y)∈Ca, i.e., y∈p−1
a(Ca) o each a∈A.
(d) The poin yis a clus e poin (in he opology T)o N.
This ollows om (24) since xd∈p−1
ad (Ua). This means ha o each
neighbo hood U∗
ao yand each b0∈A he e is a d≥b0,d∈A, such
ha xd∈U∗
a.
The p oo o Lemma 2.1 is comple e since he second s a emen easily
ollows om he defini ion o he opology Ton σ(X).
Lemma 2.2. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o compac Hausdo ff spaces. I Uis a neighbo hood o X= lim X
in σ(X), hen he e exis s a∈Asuch ha Xb⊆U o each b≥a.
P oo : Since Xis compac and since he se s (2) o m a basis o he
neighbo hoods o he poin s o X, one can find {U∗
ai:i=1,... ,n}such
ha
(25) V={U∗
ai:i=1,... ,n}
and X⊆V⊆U. In o de o comple e he p oo , i suffices o find an
a∈A,a≥a1,... ,a
nsuch ha
(26) Xa⊆V
since hen we ha e
(27) Xb⊆V⊆U, b ≥a.
Suppose ha no a∈Asa isfies (26). This means ha o each a∈A
he e is xa∈Xa−V. We ob ain a ne {xa:a∈A}in σ(X) which has
no clus e poin in V⊇X. This con adic s Lemma 2.1. The p oo is
comple e.
The F euden hal space o app oxima e sys ems 223
Lemma 2.3. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se sys-
em o compac Hausdo ff spaces. Then σ(X)is pa acompac . Mo eo e ,
i Xis an app oxima e sequence, hen σ(X)is compac .
P oo : Le V={Vµ}be any co e o σ(X). Since Xis compac , he e
is a fini e subcollec ion, consis ing o se s Vµ(1),... ,V
µ(n)which co e
X. Le Vbe he union o his subcollec ion. By i ue o Lemma 2.2
he e is an a∈Asuch ha all Xb,b≥a, a e in V. Le us ecall ha
he se X∗
a=(∪{Xb:b≥a}∪Xis o ype (2) wi h Ua=Xaand i is
open in σ(X). Now conside he ollowing collec ion Uo open se s o
σ(X): ake fi s he open se s X∗
a∩Vµ(1),... ,X∗
a∩Vµ(n) o membe s
o U. Fu he mo e, o each b∈A−{c:c∈A, c ≥a}conside he
open co e ing {Xb∩Vµ}o Xband ake membe s o a fini e subco e ing
as new membe s o U. This is possible since Xbis compac and open
in σ(X). The amily Uo open se s o σ(X) is a s a -fini e co e ing
o σ(X) which efines he co e ing V. Mo eo e , Uis a locally fini e
efinemen o V. The p oo o pa acompac ness is comple e. I Xis an
app oxima e sequence, hen we ob ain a fini e subco e ing since he se
A−{c:c∈A, c ≥a}is fini e. The p oo is comple e.
Theo em 2.4. Le X={Xn,
n,p
mn,N}be an app oxima e in e se
sequence o compac me ic spaces Xn. Then σ(X)is a compac me ic
space.
P oo : Each space Xnhas a coun able base Bn[3, 4.1.15 Theo em].
I ollows ha he amily B∗={U∗:U∈B
n:n∈N}is coun able. I
is ob ious ha he union B={Bn:n∈N}∪B
∗is a coun able base o
opology T.Thusσ(X) is me izable [3, p. 351].
We close his sec ion wi h he ollowing heo em which is simila o
he heo em o usual in e se sys ems o compac Hausdo ff spaces due
o S. Ma deˇsi´c[10, Theo em 4] (see Theo em 4.2 o [12]).
Theo em 2.5. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o compac Hausdo ff spaces and le :X→Rbe a mapping o
hei limi in o a simplicial complex. Then he e exis s an a∈Asuch
ha o each b≥aone can define a mapping b:Xb→Rwi h he
p ope y ha bpbis homo opic o .
3. Applica ions
In his sec ion we gi e some applica ions o he space σ(X). We s a
wi h
230 I. Lonˇ
ca
Lim{pa(I(y,xµ)) : a∈A}. Since each pa(I(y,xµ)) con ains I(ya,x
µ
a),
we in e ha Kµ⊆I(y,xµ). Finally, we ha e Kµ=I(y,xµ).
S ep 3.2. Fo each a∈Aand each µ∈Mwe ha e pa(Kµ)=I(ya,x
µ
a).
Clea ly, pa(Kµ)⊇I(ya,x
µ
a). Suppose ha he e is an a∈Aand a
poin za∈pa(Kµ) I(ya,x
µ
a). This means ha he e a e disjoin open
se s Uaand Vasuch ha za∈Vaand I(ya,x
µ
a)⊆Ua. F om he local
connec edness o Xai ollows ha he e is an open and connec ed se
Wasuch ha I(ya,x
µ
a)⊆ClWa⊆Ua. F om he defini ion o h ead i
ollows ha he e is a b∈Asuch ha pac(yc) and pac(xµ
c) a e in Wa o
each c≥b. This means ha pac(I(yc,x
µ
c)) ⊆ClWasince pac(I(yc,x
µ
c))
is i educible be ween pac(xc) and pac(xµ
c) (see Lemma 3.11). I ollows
ha U∗
ais a neighbo hood o a poin z∈K,pa(z)=za, such ha
U∗
a∩I(yc,x
µ
c)=∅. This means ha z/∈Ls{I(ya,x
µ
a):a∈A}=Kµ.
This is impossible since z∈Kµ. By Theo em 1.10 i ollows ha Kµ=
∩{p−1
a(I(ya,x
µ
a)) : a∈A}. Simila ly, we ha e K=∩{p−1
a(I(ya,x
a)) :
a∈A}, whe e K=Ls{I(ya,x
a):a∈A}.
S ep 3.3. Ls{Kµ:µ∈M}=Ls{I(y,xµ):µ∈M}=I(y,x).
I is ob ious ha Ls{I(y,xµ):µ∈M}⊇I(y,x) since Ls{I(y,xµ):
µ∈M}con ains xand yand I(y,x) is i educible be ween xand y.Now
we p o e ha Ls{I(y,xµ):µ∈M}⊆I(y,x). Le zbe any poin in
X−I(y,x). By i ue o he defini ion o a base in X, he e is an a∈A
such ha pa(z)=za/∈pa(I(y,x)) = (by S eps 3.1 and 3.2) I(ya,x
a).
This means ha he e is a neighbo hood Uao zaand a neighbo hood Va
o pa(I(y,x)) such ha Ua∩Va=∅. By S ep 3.2 pa(I(y,x)) = I(ya,x
a).
Since I(ya,x
a) = Lim{I(ya,x
µ
a):µ∈M}we in e ha he e is a µ0∈M
such ha , o each µ≥µ0,Uaand I(ya,x
µ
a) a e disjoin . F om 3.2
i ollows ha p−1
a(Ua) and I(y,xµ) a e disjoin . Since p−1
a(Ua)isa
neighbo hood o z, we in e ha z/∈Ls{I(y,xµ):µ∈M}. Thus,
Ls{I(y,xµ):µ∈M}=I(y,x) and 3.3 is p o ed.
S ep 3.4. I(y,x) = Lim{I(y,xµ):µ∈M}.
Apply S ep 3.3 and Lemma 3.12.
By i ue o Lemma 3.10 and S ep 3.4 i ollows ha Xis smoo h a
y. We in e ha Xis smoo h in any o i s poin y. This means ha X
is locally connec ed. The p oo o Theo em 3.13 is comple e.
A Hausdo ff con inuum Xwi h p ecisely wo nonsepa a ing poin s is
called a gene alized a c. A con inuum Xis said o be an a c i Xis a
me izable gene alized a c. A ee Xis a gene alized a c i and only i
Xis a iodic.

The F euden hal space o app oxima e sys ems 231
Theo em 3.14. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o gene alized a cs. Then X= lim Xis a iodic.
P oo : Suppose ha Tis a subcon inuum o Xwhich is a iod. This
means ha Tis he sum o h ee gene alized a cs Cx,Cy, and Cz, such
ha he common pa o each wo o hem is he common pa o all
h ee o hem and is a poin . Le x∈Cx−(Cy∪Cz), y∈Cy−(Cx∪Cz),
z∈Cz−(Cx∪Cy) and =Cx∩Cy∩Cz. By i ue o he defini ion o a
basis in X, he e exis a∈Aand open se s Vx,Vy,Vzo Xawhich a e
pai wise mu ually exclusi e and which con ain xa,ya,za, espec i ely,
so ha
p−1
a(Vx)Cy=∅=p−1
a(Vx)Cz,
p−1
a(Vy)Cx=∅=p−1
a(Vy)Cz,
p−1
a(Vz)Cy=∅=p−1
a(Vz)Cx.
Now, one o xa,yao zalies be ween aand one o xa,yao za. Suppose
ha a≺xa≺ya. Then pa(Cy) in e sec s aand yaand hence xa, bu
pa(Cy) does no in e sec Vx. This is a con adic ion. So Xcon ains no
iod.
Theo em 3.15. Le X={Xa,Ua,p
ab,A}be an app oxima e in e se
sys em o gene alized a cs wi h limi X. I he bonding mappings a e
mono one and on o, hen Xis a gene alized a c.
P oo : By i ue o Theo em 3.13 Xis a ee. F om 3.14 i ollows
ha Xis a iodic. Thus Xis a gene alized a c.
Co olla y 3.16. Le X={Xn,
n,p
mn,N}be an app oxima e in-
e se sequence o a cs and mono one on o mappings. Then X= lim X
is an a c.
P oo : Now, om 3.15, i ollows ha Xis a gene alized a c. Mo e-
o e , Xis a me izable gene alized a c. Thus, Xis an a c.
Acknowledgemen . The au ho is g a e ul o he e e ee o his help
and aluable sugges ions.
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232 I. Lonˇ
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Keywo ds. App oxima e in e se sys em and limi , F euden hal space
1991 Ma hema ics subjec classifica ions: 54B25, 54D30
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