SEMILATTICE STRUCTURES ON DENDRITIC SPACES

Vlume 2, 1977 Pages 243 260 http://tplgy.auburn.edu/tp/ SEMILATTICE STRUCTURES ON DENDRITIC SPACES by T. B. Muenzenberger and R. E. Smithsn Tplgy Prceedings Web: http://tplgy.auburn.edu/tp/ Mail: Tplgy Prceedings Department f Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: tplg@auburn.edu ISSN: 0146-4124 COPYRIGHT c by Tplgy Prceedings. All rights reserved.

TOPOLOGY PROCEEDINGS Vlume 2 1977 243 SEMILATTICE STRUCTURES ON DENDRITIC SPACES T. B. Muenzenberger and R. E. Smithsn, 1. Intrductin The algebraic structure available in dendritic spaces is refined and studied in a spirit analgus t that fund in [5], [6], and [7]. A majr cntributin is t fill a gap in the prf f a key lemma in [14] by prving that the cutpint rder gives a dendritic space the structure f a meet semilattice. The actual mechanics invlve the thery f Galis crrespndences and gap pints in partially rdered sets. The assumptin f rim finiteness n a dendritic space yields an even finer algebraic structure n the space. Fr example, it is shwn that a rim finite dendritic space has the structure f a mntne tplgical semilattice and that the tplgy n such a space is uniquely determined by the cutpint rder. The paper als includes anther cnstructin f the unique dendritic cmpactificatin f a rim finite dendritic space. Other cntributins t the thery f dendritic spaces can be fund in the wrk f Allen [1], Bennett [2], Gurin [4], Pearsn [10], Prizvlv [11] and [12], and Ward [13] and [14]. 2. Galis Crrespndences and Gap Pints in Psets Let (X,~) be a pset. Fr A c X define L(A) {x E Xla E A ~ x < a} and M(A) {x E xla E A ~ a < x}. Fr x E X define L(x) = L({x}) and M(x) = M({x}).

244 Muenzenberger and Smithsn Lemma 2.1. Let (X,~) be a pset~ Zet x E X~ and Zet A,B c X. Then a. A c LM(A) and A c ML(A). b. If A C B~ then L(B) c L(A) and M(B) c M(A). c. A c L(B) if and nzy if B c M(A). d. If A c B~ then LM(A) c LM(B) and ML(A) c ML(B). e. LML(A) = L(A) and MLM(A) = M(A). f LM(x) = L(x) and ML(x) M(x). g. If x E L(A)~ then L(x) c L(A). If x E M(A)~ then M(x) c M(A). As bserved in [3] and [9], statements (a) and (b) taken tgether say that Land Mare GaZis crrespndences fr the subsets f X. Statement (c) is an equivalent frmulatin due t J. Schmidt [3, page 124]. Statements (d) and (e) are prved in [3] and [9]. Statement (f) is alluded t n page 126 f [3], and statement (g) fllws easily frm transitivity. If (X,~) is a pset and X c X, then x E X is a gap pint f X if and nly if (L(x) - {x}) n ML(X ) = ~. Lemma 2.2. Let (X,~) be a pset and Zet X c X. Then a. If x E X n L(X ) ~ then x is a gap pint f X 0 0 b. N tw gap pints f X are cmparabze. c. If x is a gap pint f X and x E Y c ML (X ) ~ then x is a gap pint f Y. Prf. (a) If x E X n L(X )' then ML(X ) = ML(x) = M(x) and s x is a gap pint f X. (b) If x and x' are gap pints f X with x < x', then x E (L(x') - {x'}) n ML(X ) =~.

TOPOLOGY PROCEEDINGS Vlume 2 1977 245 (c) If Y C ML (X ), then ML (Y) c: ML (X ) frm which the result fllws readily. Lemma 2.3. Let (X,~) be a pset and let X c X be a nndegenerate set f nncmparable elements. a. X n L(X ) = ~. Then b. If x is a gap pint f X and L(x) is a tset, then L(X ) = L(x) - {x}. Prf. (a) An element f X n L(X ) wuld be cmparable t any element f X. (b) L(X ) c L(x) {x} since x E X L(X ). Suppse that y < x. Since x is a gap pint f X ' there exists a E L(X ) such that a t y. S y < a and thus y E L (X ) 3. Mds Let (X,~) be a pset and fr x,y E X define x A y inf {x,y} when the infimum invlved exists. Definitin 3.1. A pset (X,~) is a md if and nly if the fllwing cnditins hld: a. Fr all x,y E X, X A Y exists. b. Fr all x E X, L(x) is a tset. c. Each nnempty subset f X which is bunded abve (belw) has a supremum (an infimum) in X. d. If x,y E X and x < y, then there exists z E X such that x < z < y. S a md is a cnditinally cmplete and rder dense meet semilattice in which the lwer sets are tsets. There is an alternative apprach. Let X be a set, let P be a cllectin f subsets f X, and cnsider the fllwing

246 Muenzenberger and Smithsn five axims n (X,~). x,y E P. Axim 1. If x,y E X~ then there exists P E ~ such. that Axim 2. If ~ ~ ~ C P and if n ~ ~ ~~ then n ~ E~. 000 If x,y E X, then the chain with endpints x and y is defined by [x,y] = n{p E ~Ix,y E pl. Set [x,y) = [x,y] - {y} = (y,x] and (x,y) = [x,y] - {x,y}. A set A c X is chainable if and nly if x,y E A imply [x,y] c A. Axim 3. If P E ~~ then there exist unique x,y E X such that P = [x,y]. Axim 4. The unin f tw chains that meet is chainable. Axim 5. If x,y E X and x ~ y~ then (x,y) ~ ~. If e is an element f X, then the chain rder < with basepint e is defined by the rule: If x,y E X, then x < y if and nly if x E [e,y]. Therem 3.2. If (X,~) satisfies Axims 1-5~ if e E X~ and if ~ is the chain rder with basepint e~ then (X,~) is a md with least element e and the chains bey the fllwing equality: =1L( ) n M(x) if x ~ y. [x,y] [XI\Y, x] U [XAY, y] A prf f Therem 3.2 can be fund in [5] if x and y are nt cmparable. Therem 3.3. If (X,~) is a md and the chains in ~ are defined as in the previus therem, then (x,~) satisfies

TOPOLOGY PROCEEDINGS Vlume 2 1977 247 Axims 1-5. Mrever~ if f E X~ then the chain rder with basepint f is exactly (~-{(x,y) Exxxlx<fAy}) U{(x,y) Exxxif Ay~x<f}. A prf f the first part f Therem 3.3 may be fund in [5]. The secnd part f Therem 3.3 is nt necessary fr the primary purpses f this paper, and s the prf is mitted. Therems 3.2 and 3.3 shw that the cncept f a md is cextensive with that f a md with least element. Many results prved in [6] did nt require the existence f maximal elements, and s they are valid in the mre general cntext f a md. Here is an example. An arc is a Hausdrff cntinuum with exactly tw nncutpints. A Hausdrff space X is acyclic if and nly if any tw distinct pints x,y E X are the endpints f a unique arc A[x,y] in x. An acyclic space admits a natural md structure which will nw be described. Define A[x,x] = {x} where x E X. If X is an acyclic space and e E X, then the arc rder < with basepint e is defined by the rule: If x,y E X, then x ~ y if and nly if x E A[e,y]. It was shwn in [8] that (X,~) is a md. In fact, there is a tplgical characterizatin f mds with certain natural rder cmpatible tplgies in terms f acyclic spaces which is analgus t Therems 7.1 and 7.2 in [6]. If (X,~) is a md with least element and each nnempty tset in X has a supremum in X, then (X,~) is called a semitree. Semitrees were studied extensively in [5]-[7]. 4. Dendritic Spaces Let X be a cnnected Hausdrff space. If e is an

248 Muenzenberger and Smithsn element f X, then the cutpint rder ~ with basepint e is defined by the rule: If x,y E X, then x ~ y if and nly if x = e, x = y, r x separates e and y. Let Max(X) dente the < maximal elements f X and let N(X) dente the nncutpints f X. Lemma 4.1. Let X be a cnnected Hausdrff space 3 let e E X 3 and let < be the cutpint rder with basepint e. Then a. ~ is a partial rder with least element e. b. Fr all x E X 3 L(x) is a tset. c. Fr all x E X 3 M(x) - {x} is an pen set. d. If x,y,z E X and z separates x and Y3 then z < x r z < y. e. If X is nndegenerate 3 then Max(X) = N(X) - {e}. Prf. statements (a)-(c) are prved in [14]. (d) Suppse that X - {z} = A U B where x E A, Y E B, and A and B are separated sets. If z = e, then z < x,y. If z ~ e, then z < x r z < y depending n whether e E B r e E A. (e) See the prf f Therem 13 in [14]. A dendritic space is a cnnected Hausdrff space in which each pair f distinct pints can be separated by sme third pint. Thrugh Prpsitin 4.7 in this sectin, X will dente a dendritic space, e will dente an element f X, and < will dente the cutpint rder with basepint e. The fllwing Lemma is als prved in [14]. Lemma 4.2. (aj The partia l rder 2. is dense. (b J Fr all x E X 3 L(x) and M(x) are lsed sets. (cj The partial

TOPOLOGY PROCEEDINGS Vlume 2 1977 249 rder < is a clsed subset f X x X. The remainder f what is needed t shw that (X,<) is a md wuld seem t be prvided by Lemma 8.2 f [14]. Unfrtunately, there is a gap in the prf f Lemma 8.2 given therein because, in the ntatin f the prf f that lemma, Xl and y need nt be cmparable. In effect, what is needed t cmplete the prf f the Lemma is the existence f xl A y, and the main part f this sectin will be devted t prving the existence f the infimum f a dubletn in X. Lemma 4.3. If X c X is a set f nncmparable elem~nts, then there exists at mst ne gap pint f X. Prf. Suppse that xl and x are distinct gap pints 2 f X. By Lemma 2.3, [e,x 1 ) = L(X ) = [e,x ). Let z E X 2 separate xl and x 2. By Lemma 4.1, it may be assumed that z < xl. Then z i ML(X ) since xl is a gap pint f X. S there exists a E L(X ) such that a z. But a,z < xl whence z < a since L(X ) is a tset. Nw there are tw separatins 1 f interest here. First, X - {z} = C U D where xl E C, t x 2 ED, and C and D are separated sets. Secnd. X {a} (X - M(a) ) U (M(a) a) where e,z E X - M(a), xl and x are in M(a) - {a}, and 2 X - M(a) and M(a) - {a} are disjint pen sets. The questin is where des a lie in the first separatin? If a E C, then D U {z} is cnnected and meets bth pieces f the secnd separatin. Similarly, a E D leads t a cntradictin. S X has at mst ne gap pint. Crllary 4.4. If X c X, then there exists at mst ne

250 Muenzenberger and Smithsn gap pint f x Prf. Suppse that xl and x are distinct gap pints 2 f X. Then xl and x are gap pints f Y = {x l,x } c X 2 2 by Lemma 2.2(c). By Lemma 2.2(b), xl and x are nncmpara 2 Therem 4.5. If x l,x E X 3 then xl A x exists. 2 2 Prf. Let x l,x E X and define B = ML{x,x }. Observe 2 l 2 that ML(B) = B by Lemma 2.l(e), that B = n{m(a) la E L{x l,x }} 2 is a clsed set by Lemma 4.2(b), and that B has at mst ne gap pint by Crllary 4.4. Suppse first that B has n gap pints. Then (1) B = U{M(b) - {b} Ib E B}. Fr suppse that X E B. Then [e,x) n B ~ ~ since x is nt a gap pint f B, and if b E [e,x) n B, then x E M(b) - {b} where b E B. On the ther hand, if x E M(b) - {b} where b E B, then x E M(b) c B by Lemma 2.l(g). By (1) and Lemma 4.I(c), B is an pen set. Thus, B is a clpen set, and therefre B = X. S e E B, but e is a gap pint f X by Lemma 2.2(a). Thus, B must have exactly ne gap pint. ble. Then Lemma 4.3, when applied t Y' prvides a cntradictin. Suppse that b is the nly gap pint f B. Then (2) B - {b } = U{M(b) - {b} Ib E B}. The prf f (2) is identical with the prf f (1) save fr the bservatin that b i M(b) - {b} when b E B because [e,b ) n B =~. S B - {b } is an pen set. If e E B, then 0 e = xl A x 2. Suppse that e i B. If xl = x 2 ' then xl = xl A X 2 = x 2. Suppse that xl ~ x 2. Then

TOPOLOGY PROCEEDINGS Vlume 2 1977 251 (3) X - {b } = (X - B) U (B - {b }) 0 0 is a separatin. If b = xl' then (3) implies that xl < x 2 and xl /\ Similarly, b x implies that x = = xl x 2 0 2 2 xl /\ x 2 Finally, if xl ~ b ~ then (3) implies that x 2 ' Crllary 4.6. The pair (X,~) is a md. Prf. There are tw avenues f attack here. First, it can be prven directly frm Therem 4.5 that each nnempty subset f X has an infimum in X, and frm that it can be easily prven that each nnempty subset f X which is bunded abve has a supremum in X. Secnd, the gap in the prf f Lemma 8.2 f [14] has nw been filled, and s that Lemma may als be applied t yield the desired cnclusin. If X is a dendritic space and X c X, then inf X is a gap pint f ML(X ) and vice versa. O In fact, the cncepts f gap pint and infimum can be made t cincide in a dendritic space by changing the definitin f gap pint t nly require that a gap pint X f X be a member f ML(X ) instead f re O quiring that x lie in X itself. Therem 4.5 als prvides simple prfs f several resuits in [14]. Here is an example. Prpsitin 4.7. Fr all x E X~ L(x) is clsed. Prf. Let yi L(x) and chse t s that x /\y < t < y. Then y E M(t) - {t} c X - L(x). In fact, fr any x E X X - L(x) = U{M(t) - {t}lt E X and t f x}. Fr anther applicatin, nte that the existence f md

252 Muenzenberger and Smithsn structures n acyclic and dendritic spaces means that the fixed pint therems prven in [5], [6], [8], and [14] are valid fr such spaces. The final therem in this sectin can be prved by using Therem 3.3, Lemmas 4.1 and 4.2, and Prpsitin 2.3 in [14]. Therem 4.8. Let X be an acyclic r a dendritic space~ let e E X~ and let < be the arc rder r the cutpint rder with basepint e~ respectively. Then the family P f chains described in Therem 3.3 is independent f e. In fact~ if f E X~ then the chain rder with basepint f determined by P equals the arc rder r the cutpint rder with basepint f~ respectively. 5. Rim Finite Dendritic Spaces If A is a subset f a space X, then A will dente the clsure f A in X. A space X is rim finite if and nly if each element f X admits arbitrarily small neighbrhds whse bundaries are finite. Thrugh Therem 5.11, let X be a rim finite dendritic space and let X* dente the unique dendritic cmpactificatin f X first given by Prizv1v [11] and later refined by Allen [1], Pearsn [10], and Ward [14]. By the cnstructin f X* given in [14], x* = X U N where X n N = ~ and N C N{X*). This is because the cutpints f X* are all members f X as bserved in the prf f Therem 23 in [14]. Let e E X, let < be the cutpint rder n X with least element e, and let ~* be the cutpint rder n X* with least element e. Let A and A* be the crrespnding infimum peratins. By Crr1 1ary 4.6, (X'~) and (X*'~*) are mds. Since X* is a tree

TOPOLOGY PROCEEDINGS Vlume 2 1977 253 (a cmpact dendritic space), (X*'~*) is actually a semitree, and J*, the tplgy f X*, is strngly rder cmpatible as defined in [6]. The later fact is well knwn, and indeed it fllws frm Lemmas 4.1 and 4.2, Therem 4.8, and certain results in [6]. Thus, if x,y E X* and [x,y]* dentes the chain in x* with endpints x and y, then [x,y]* is the unique arc in X* with endpints x and y. Furthermre, the family cnsisting f all sets f the frm M*(x) - {x}, X* - M*(x), X* - L*(x) where x E X*, L*(x) {y E x*ly ~* x}, and M*(x) = {y E x*lx ~* y} is a subbasis fr ]* by Therem 4.27 in [6]. The purpse f the present sectin is t extend these and ther results t X in s far as pssible. Lemma 5.1. If m E Max(X*), then the family ] = {M*(t) - {t}lt E X* and t <* m} is a basis fr the neighbrhds f m in x*. Prf. Let x E X* and m E X* - M*(x). If x A* m <* t <* m, then m E M*(t) - {t} c X* - M*(x). The lwer sets have already been handled in Prpsitin 4.7. Furthermre,] is clsed under finite intersectins, and s ] basis at m in x*. is an pen Crllary 5.2. If x E X*, then a separatin X* - {x} CUD f x* - {x} restricts t a separatin X - {x} = (c n X) U (D n X) f X - {x}. Prf. If C n X = ~, then C C N c Max(X*). But C is pen, cntradicting Lemma 5.1 r else the lcal cnnectivity f X*.

254 Muenzenberger and Smithsn Lemma 5.3. The inclusin <* n (X x X) c < hlds. Prf. Apply Crllary 5.2. Actually, Crllary 5.2 is nt necessary t prve Lemma 5.3, but Crllary 5.2 will be needed later in the prf f Crllary 5.8. Lemma 5.4. If C cx* is chainable and M cmax(x*), then C - M is chainable and therefre cnnected. Prf. If x,y E X* - M, then [x,y] * c L* (x) U L* (y) c X* - M. Hence, C - M is chainable and therefre cnnected by Therem 4.21 in [6]. Crllary 5.5. Max(X*) = Max(X) UN. Prf. Apply Lemmas 4.l(e), 4.2(a), 5.3, and 5.4. All clsures in the next Lemma and prf are t be taken in X*. Lemma 5.6. If x E X, then a separatin X - {x} = CUD f X - {x} extends t a separatin X* - {x} = (C - {x}) U (0 - {x}) f x* - {x}. Prf. Ntice that end = {x} by Lemma 1.2 in [1], and s C - {x} and IT - {x} are separated sets in X* - {x}. Nw X - {x} = X*, and s X* - {x} = (C - {x}) U (0 - {x}) is a separatin. Crllary 5.7. The partial rder <* restricted t X x X equals ~. Of curse Crllary 5.7 fllws immediately frm Lemmas 5.3 and 5.6, but alternate prfs are pssible by verifying the cnditins in Therem 5 r Crllary 11.1 r 12.1 in [14].

TOPOLOGY PROCEEDINGS Vlume 2 1977 255 Crllary 5.8. N{X*) = N{X) UN. Prf. Lemmas 4.l{e) and 5.6 and Crllary 5.5 are enugh t shw that N{X*) c N{X) U N' and Crllary 5.2 shws that N(X) cn{x*). Crllary 5.9. The equatin A*l = A hlds. xxx Prf. This is immediate frm Crllary 5.7. Therem 5.10. The pair (X,A) is a mntne tplgical semilattice. Prf. The functin A* is cntinuus since X* is a tree. S A is cntinuus by Crllary 5.9, and A is shwn t be.mntne in the prf f Therem 11 in [7]. Examples 3 and 4 in [7] shw that an arbitrary dendritic space need nt have a cntinuus infimum peratin. Hwever, Therem 11 in [7] des yield a result abut the cntinuity f the infimum peratin n certain dendritic spaces that is slightly better than the ne in Therem 5.10. T see that it is a better result necessitates a fuller illuminatin f the structure f a rim finite dendritic space. It seems t be well knwn that a rim finite dendritic space X is acyclic; see, fr example, [2], [4], [10], [11], and [14]. In fact, if x,y E X, then [x,y] = [x,y]* is the unique arc jining x and y by Lemma 5.7, Crllary 5.9, and the crrespnding structure n X*. Thus, the cutpint rder n X with basepint e cincides with the arc rder n X with basepint e. Cmpare Therem 4.8. Therem 5.11. The family cnsisting f all sets f the frm

256 Muenzenberger and Smithsn M(x) - {x}, X - M(x), X - L(x) where x E X is a subbasis fr the tpzgy J f x. Prf. The subbasis fr J* described at the beginning f this sectin restricts t a subbasis fr J. Hwever, fr x E X* (M* (x) - {x}) n X =i:(x) (X* - M*(x)) n =ix-m(xl X X - {x} = X - {x} if x if x if x if x E X. rj X. E X. i X. (X* - L* (x)) n X =ix L(xl if x E x. U{M(t) - {t} It E X, t 1.* x} if xi x. The last equality hlds because f the equality fllwing Prpsitin 4.7. S the family described in the statement f the therem is als a subbasis fr J. Let (X,J) be a cnnected space with tplgy J. Fllwing Ward [14], let a dente the tplgy generated by the family f all cmpnents f sets f the frm X - {x} where x E X. Suppse nw that (x,~) is a md and let J dente 0 the tplgy generated by the family f all sets f the frm The tplgy J H(x) - {x}, X - M(x), and X - L(x) where x E X. 0 is called the tree tp lgy determined by < Crllary 5.12. If (X,J) is a dendritic space~ if e E X~ and if ~ is the cutpint rder with basepint e~ then the fllwing statements are equivalent. a. (X,J) is rim finite. b. J a. c. J J ~ the tree tplgy determined by ~. Prf. (a) and (b) were shwn t be equivalent in

TOPOLOGY PROCEEDINGS Vlume 2 1977 257 Therem 21 in [14]. (a) =;. (c) is Therem 5.11. (c) 9 (b) Observe that the cmpnents f X - {x} where x E X cnsist f X - M(x) and all B - {x} where B is a branch at x as defined in [6]. Apply Lemma 4.19 in [6] and the equality fllwing Prpsitin 4.7. Several ther cnditins which are equivalent t rim finiteness in a dendritic space are given in [14]. Therem 5.13. Let (X,~) be a md with least element e and let J be the tree tplgy determined by ~. Then J is a strngly rder cmpatible tplgy and (X,J ) is a rim finite dendritic space. Mrever~ J is the nly rim finite dendritic rder cmpatable tplgy n x. Prf. Let? dente the family f chains in X, let J(?) dente the strng tplgy n X induced by?, and let!) dente the interval tplgy n X (see [6]) Then!) cj c J(?) by definitin f J and Lemma 4.3 in [6] 0 0 Hence, J is a strngly rder cmpatible tplgy. Nw (X,J ) is dendritic by Lemmas 4.19 and 4.25 and Therem 4.21 in [6], and (X,J ) is easily seem t be rim finite. Let J be an rder cmpatible tplgy n X fr which (X,J) is a rim finite dendritic space. Then fr x,y E X, the chain [x,y] is an arc in (X,J), whereas the space (X,J) is acyclic as bserved earlier. Accrdingly, the cutpint rder n (X,J) with basepint e, the arc rder n (X,J) with basepint e, and the chain rder n (X,?) with basepint e (namely, ~) all cincide. Crllary 5.12 then implies that J J S there d nt exist tw nnhmemrphic rim finite

258 Muenzenberger and Smithsn dendritic spaces having rder ismrphic md structures cmpatible with their respective tplgies. When applied t an acyclic space with an arc rder, Therem 5.13 shws that the rim finite dendritic spaces play the same rle amngst the acyclic spaces that the trees play amngst the nested spaces [6] There is anther cnstructin f X* which is identical as a set t that prvided by Ward in [14], but which differs frm his cnstructin in that the tplgy is derived frm the rder. Let (X,~ be a rim finite dendritic space, let e E X, and let < be the cutpint rder with basepint e. Then (X'2) is a md and] = ]0 is the tree tplgy determined by <. Fr each maximal < tset T in X with n largest element, adjin t X an element x i X. Let N T a maximal < tset in X with n largest element} and let X* = X U N. Extend < t X* as fllws. a. If x,y E X, then x <* y if and nly if x < y. - - b. If x,y E N 0' then x - <* y if and nly if x = y. c. If x E X and y E N then x <* y if and nly if 0' x E T, a maximal < tset in X with n largest element, and y = x T. Then (X*,~*) is a semitree and <* n (X x X) = <. If ]* is the tree tplgy determined by 2*' then (X*,]~) is cmpact Hausdrff and dendritic by Therem 4.27 in [6]. It can be shwn that J; n X is a rim finite dendritic rder cmpatible tplgy n X, and hence, ]~ n X = J by Therem 5.13. But the later equality is mre readily derivable frm the equalities listed in the prf f Therem 5.11, and thse equalities are easily shwn t be valid in the present cntext.

TOPOLOGY PROCEEDINGS Vlume 2 1977 259 Observe that the cnstructin f X* given herein really des depend n the cnstructin prvided by Ward in [14] since Therem 5.11 required the existence f X* in its prf and was used in bserving that J = J. References [1] K. R. Allen, Dendritic cmpactificatins, Pacific J. Math. 57 (1975), 1-10. [2] R. Bennett, Peripherally cmpact tree-like spaces are cntinuum-wise cnnected, Prc. Anler. Math. Sc. 39 (1973), 441-442. [3] G. Birkhff, Lattice thery, American Mathematical Sciety Cllquium Publicatins, Vlume 25, Prvidence, 1967. [4] G. L. Gurin, Tree-like spaces, Vestnik Mskv 24 (1969), 9-12. [5] T. B. Muenzenberger and R. E. Smithsn, Fixed pint structures, Trans. Amer. Math. Sc. 184 (1973), 153-173. [6], The structure f nested spaces, Trans. Anler. Math. Sc. 201 (1975), 57-87. [7], Mbs and semitrees, J. Austral. Math. Sc. 21 (1976), 56-63. [8], Fixed pint therems fr acyclic and dendritic spaces, Pacific J. Math. 72 (1977), 501-512. [9]. are, Thery f graphs, American Mathematical Sciety Cllquium Publicatins, Vlume 38, Prvidence, 1962. [10] B. J. Pearsn, Dendritic cmpactificatins f certain dendritic spaces, Pacific J. Math. 47 (1973), 229-332. [11] V. V. Prizv10v, On peripherally bicmpact tree-like spaces, Sviet Math. Dk1. 10 (1969), 1491-1493. [12], On the hereditary and cllective nrmality f a peripherally cmpact tree-like space, Sviet Math. Dk1. 11 (1970), 1072-1075. [13] L. E. Ward, Jr., On dendritic sets, Duke Math. J. 25 (1958), 505-514.

260 Muenzenberger and Smithsn [14], Recent devezpments in dendritic spaces and rezated tpics, Studies in Tplgy, 601-647, Academic Press, New Yrk, 1975. Kansas State University Manhattan, Kansas 66506 and University f Wyming Laramie, Wyming 82071