An arc problem

23 Pages·01.274 MB·English

by  Boyer, Jean Marie

m Mm PBOQUSU
By
dean II* Boyer
Thesis submitted to the Faculty of the Graduate School of
the U niversity of Maryland in p a rtia l
fulfillm ent of the requirements for the
degree of Doctor of Philosophy
19 SI
UMI Number: DP70277
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The w riter wishes to express her sincere
appreciation to Professor Dick wick Hall for
suggesting th is problem and for his patient
and generous expenditure of time arid effo rt
during the preparation of th is thesis*
m me PROBLEM
A topological apace A is said to be embedded topologically in a
topological space B if there exists a subset A* of the space B and a
transform ation T(A) * A* which is one-to-one and continuous in both
directions# Under these conditions the set A* is said to be hcmeomor~
phie to A, and the transf orsa&t i on is called a home ©morphism*
The question as to whether a given topological space A can be
embedded topologically in a topological space B is an unsolved and
apparently extremely d iffic u lt problem* I t may be approached fro® at
least two points of view* I' irs t, one may require that there ex ist a
subset A1 of B and a single valued continuous mapping T(A) * A*, and
then seek conditions on th is mapping T to insure that i t is a homao-
Morphism. Such an approach has been used by J. f # l&rdwell*^ I t can be
considered an analytic approach to the problem*
The other approach {the one which w ill Interest us here) attem pts
to solve the problem from a structure-theoretic standpoint. In other
words, one attem pts to solve the problem by placing additional hypotheses
on the structure of the space A rather than on a mapping fro® th is
space into the space B*
The problem can be wade more meaningful, perhaps, if we re stric t
F* Aardwell, ^Continuous transform ations preserving a ll topological
properties", im & .cm .Jm m A . «&Jfctfeeasagft. rol.S8 (1236), pp. 709-726.
z
the spaces A and B in our discussion to spaces having w ell known
properties. Consider, for example, the case where A is an arbitrary
compact locally connected continuum (a Peano space) and where B is
the two dimensional sphere * 1. The set A is said to
b@ skew if it cannot be embedded, topologically in the set B*
Kuraiowski has introduced the following two sets, which have
bee os© quite famous In topology*^
A primi t ive skew curve is said to be of type 1 If it consists'
of six d istin ct points P^, Pg, Fg, q^, q^, qg, and. nine arcs
• •• , with end points as indicated and with the common
part of two of these arcs that in tersect each other being an end
point of each*
A prim itive skew curve is said to be of type 2 if it consists
of five d istin ct points P^, P^, Pg, P^, Pg, and ten ©res P^Pg,
V § , , P^Pg with and points as indicated and with the common
part of two of these arcs th at in tersect each other being an end
point of each*
luratowski ha© proved that a skew Peano space containing only
a fin ite number of simple closed curves most contain a prim itive
skew curve of type 1 or type 2.^ Claytor showed, that a cyclic
Peano space can be skew only if i t contains one of these two types
S?
Cm luratow ski, "Sur la problem® des courbes gauches en
topologie*, i undamenta Mathematics® * vol* 15 (1950), pp* 271-285*
5
Euratowski, loc* c it*
s
4
of curves#
g
R# H# Bing has recently stated the following two theorems#
The proofs of these theorem® were not published because of th eir
lengths#
SaarauM Zmssss. .t-fcsfe px* pa. ps» -*x« *5 «sa j^s. <!£*-
tiJBsS; gaM a sM is a l px%i* pi^2> *** * ps*s aia s&ai. *><>» »*•**>
^ jglntg as jy^LcjrbsS ang. jgug. k iu i &m °L tb*«* area intwaact
SSSfe 2k£S£ SSiX i£ th#y. t e a fiQ m l aslnl la eqwon. Then tfc, jggg
J°£ IMS® arffig Is ask bpisaomoruhlc £o J2lgQA 3*i«
Theorejajit ffuppogg tltSfc i*x» PE* p8* p4> PS ^£2. £*■** distinct
ao^nts and that PxP2, P-jPg, ... , P4Pg aEg t^n arcs g&kfe £2*.
M ints as indlcatad and sucfo that two of thasa arcs intargact
*M i akfeg£ ,snjx 1£ they iifiZE £& S2& P.°<wt la oseESB* Tiw>n tiifi auj£
of thfiSg argg Is ask ftBMBMEBte. £a £££ 2l«Z2a £S&‘
The reader is cautioned to note carefully the difference
between the set described in Theorem A and a prim itive skew curve
of type 1| also the difference between the set described in Theorem
B and a prim itive skew curve of type Z* It is evident th at either
of these theorems could be proved by constructing in the set
described a prim itive skew curve of either type 1 or type 2#
The discussion of the previous paragraph indicates, and i t
can be proved rather easily , that both Theorem A and Theorem B
4
£»• Claytor, ^Topological immersion of Peanian continue in
a spherical surface11, Annals of Mathematics, vol# 55 (1954), pp#
809-055#
5
E# E* Bing, **Sk@w Sots1*, American Journal of laathematice*
vol* 69 (1947), pp. 495—498#
4
would follow a t one© if i t knmm that the following sore
6
general Huesiion of Bing could bo anm®r®& in the affincatlvttt
•twesMoni I . tba following utatesont fcrua? I f n^. ag, ... ,
a ara area two of ir^cU intorwotL or.ly if th«y aav-j an end aolwt
4n cordon, then ther, sxtrt M ££ ?x* P2> ••• * Pn &'<&& Saak
i) km a£ ktesst s m . intoreoot $ . ax a m M m m
12
end point ia coiamo^i
ii} 4M «■«*» jesEfe £L k*a M ast &aa JXak
ia connaetadt
ill) {L (i * ia an arc in U a 4 having m m
/ =i
nn4 ittrijaaa m «t*
In tiiia paper w» answer th is question In the affirm ative in
each of the following' easee*
a) At least i*-2 of the arcs neve a common and point*
b) $© three ares hmm a coas&on arid pointy if n is even*
c) the integer n is lee® than five.
B® also establish the tru th of the statement In certain
special cams when n * 5,6,
I t is found in the proof th at, for small values of n a t le a st,
the problem ©ay be subdivided into the consideration of certain
simple geoEsetrie eonfl&uations« It ia hoped that the suet hod® devel­
oped here can be extended eventually to answer the general question
in the affirmative*
%±ng, ley* c it* p
5
By modifying the conclusion of the stateisent slig h tly w©
obtain a the ore© valid for a ll values of n which is easily proved#
This f.eneral resu lt is stated as the fir s t theorem of the paper#
Throughout the paper the notation x » (rssK) w ill bo used to
denote the f ir s t point of the closed set K on the simple arc r® in
the order from r to s. The end points of the are w ill be
denoted by b^#
The following theorem differ® from the problem of Bing in
th at two intersecting arcs p^, are not required to have an
end point in common#
UL «!» °j}» ••• > «„ 3ES SE£S is £ M. s&s&
m ix. U they have an end point in cpBff.nn. then ttmra gxjgt area
h » p£. • •• > Pn SSSM iiiSi
a) t£g. cogr on sarfc s& iK2 fi£ thaw acga ia Sg?,tWPi<Kl
SC emetyt
A*
b) Pi (i * 1 ,2 ,. ...n ) i£ . £££. ic U a hgftSg th& £§Bfc
32
/*« J
S2& 22tete ££. Oil
c) set b_ - j p c_c«it&in« m Mends. cisssA s m a i
“ *•-( 1
Proof* The proof w ill be by Induction on the number of are®*
The theorem is obviously true for n » 1. Se shall assume it is true
fear n m k-1 and prove i t is true fo r n • k#
Choose any k-1 of the given arcs as a ., a,
can find ares p^> ?g* ••* i satisfying the required conditions#
K'l
^ t B|c^ * p^. Every component of which in tersects
contains at least one of the point© a^, b^« Thu® there exist a t most
6
two such components*
If and ^ / ^k-X ®k and are disjoint* Thus
w© can define p^ * o^*
If ^ e Bk_^, and bk / B ^ , le t p - ( b ^ s B ^ ) . Then ftj, and
p must lie in the sac* opponent 8* of There is an arc y in
B1 from to p since B* is ercwis© connected* Define *"
( f) U (pbk of ak).
If a^ and b^ are in the mam component B» of there is an
arc y in B1 fro® to bfe* Define p^ • y*
If and b^ are in d istin ct components A* and B * respectively
of 1st <4 «* (b^a^fA*) ami le t r «• (qb^fB1)* Then there is aa
arc Tj in A* fro* ^ tc % and an arc r2 in B* from r to bfe. Befin.
pk ■ (ri> U (qr of sk) U (r2).
The reader may note that if p^ f\ p^ is not connected for soma
i * l #2#***jte»l then there is a subarc a of p^ spasming p^ and
7
lying- in 8^ ^* Hence the union of p^ and a contains a simple closed
curve* This contradicts the inductive hypothesis*
K
If we define B * \J p^ i t is easy to see th at contains
^ - /
no simple closed curve, and the arc® p^ (i * l,2 f«*«9k) satisfy the
required conditions* Bens© Theores 1 1© established*
An example w ill show that th is thefcro® is not true if th©
additional restrictio n is imposed that two intersecting arcs p , Pj
k nozkdegenerete arc c span® an arc 5 if a has exactly its end
points on 6* An arc cr with d istin ct end points r,s is ©aid to span
f ran a set A to a set B provided r e A, © s Bf and (o-r-s) C\ (A U B) » 0*
7
have a common end point* Lot
<»i f) \ - 8^*!* ( i * 1»2»3),
“ b4 - * r
To satisfy the additional restrictio n we mustc haos® p ® a^9
(1 • 1,2,3,4}* This choice gives us & staple closed curve in
•4
{J fi. which contradicts condition c)•
• •/ * *
In discussing' the problem in it® original form wo shell ca ll
/n /•**
n the order of the set U «.* He can assume U <%. is connected,
X-' 1 ;r/ 3.
since If not, the problem can be reduced to several cases of lower
order* Also, if * a^ and » bj for any k ,j we can- choose
pfc * pj since 11) and ill) are obviously satisfied and may
in tersect anything that p^ is permitted, to intersect* Thus we can
assume m and. reduce th© order of the set by one* We shall
henceforth consider the order of our cot® ha,® been reduced as
much a© possible in th is manner*
For n » 1 the solution to the problem is triv ia lly in th®
affirm ative* Hence we can sesum n ^ 1, and need not establish a
p articu lar ease In each inductive argument*
thPWMi .gt The lias sea kt aanesssi 4a
wa&Am s£ itm.
atflaM a sate mis mm. 2f. %M £ails4E£ ss
a) All the arcs have a common end point* In th is case
we oan number th© arcs in any order and so choose p^, Cl * l,£,***,n)
th at U 0 contains no simple closed curve* Moreover, 04 c: \J a4
1 - / 1 J 4 r /
£or j * l,2,.**,n *
b) Exactly n-1 of the arcs have a common end point* In
rtv
th is case \J contain® at most one simple closed curve*
i-zl