Research Announcement: MAXIMAL CONNECTED HAUSDORFF TOPOLOGIES

Transcription

1 Volume 2, 1977 Pges Reserch Announcement: MAXIMAL CONNECTED HAUSDORFF TOPOLOGIES by J. A. Guthrie, H. E. Stone, nd M. L. Wge Topology Proceedings Web: Mil: Topology Proceedings Deprtment of Mthemtics & Sttistics Auburn University, Albm 36849, USA E-mil: ISSN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume Reserch Announcement 349 MAXIMAL CONNECTED HAUSDORFF T,OPOLOGIES J. A. Guthrie, B.E. Stone, nd M.L. Wge 1. One of the more interesting problems in the lttice of topologies during the lst ten yers hs been the question [4,3] of the existence of non-trivil Husdorff topologies mximl with respect to connectedness. Properties of such spces nd prtil results hve been obtined by Thoms [5], Reynolds [1], nd Guthrie nd Stone [1,2] mong others. The object of this nnouncement is to outline solution by constructing mximl connected expnsion of the rel line. Such constructions hve been obtined jointly by Guthrie nd Stone, nd independently by Wge. Both will be described briefly, nd they will be compred, but detils will pper elsewhere. Both exmples re modifictions of the rel line R with the Eucliden topology E. We conclude with some discussion of the relted open questions which remin. 2. It hs long been known tht expnsions of connected topologies obtined by djoining filter of dense sets remin connected; hence necessry condition for spce to be mximlly connected is tht the spce be submximz--every dense set is open. In ttempting to get mximl connected spce, it is nturl to djoin sets s fr removed from dense sets s possible--in fct, to exmine expnsions which to not chnge the dense sets. A typicl such set is the inverse imge of n intervl bout 0 under the topologist's sine curve. In prticulr, we cll set S c R singuzr t xes if

3 350 Guthrie, Stone, nd Wge ) S - {x} E E b) x E CI (S n (_oo,x)) n CI (S n (x,oo)). E E We cll n expnsion of E singulr expnsion if every locl bse t x consists of sets singulr t x. set is open, we cll the spce nonsingulr. If every singulr Since clerly djoining singulr set preserves connectedness, nonsingulrity is necessry condition for mximl connectedness. Theorem 1. An expnsion of the rels which is connected, nonsingulr nd submximl is mximlly connected. It is esy to get nonsingulr expnsion of E: merely djoin for ech x mximl filter of sets which re singulr t x. The problem is to obtin such spce which is connected. The key to such construction is the following result bout disconnected singulr expnsions of I = [0,1]. Theorem 2. Let (A,B) disconnect singulr expnsion of (I,E). If C = I - (IntEA U Int B), then (C,E) is the Cntor spce. Ech disconnection (A,B) is relted to Cntor set; but there re t most 2 w of these. Hence for ech rel x we cn ssocite copy of the Cntor set C(Ax,B ) such tht x x is non-endpoint of C, nd set Sx singulr t x such tht Sx n Ax nd Sx n B both cluster to x. Refine {Sx} to x mximl singulr filter t x nd djoin ll such to E. The result must be connected, for ech possible disconnection hs been defeted in dvnce.

4 TOPOLOGY PROCEEDINGS Volume Theorem 3. There exists connected nonsingulr expnsion of E; hence there is mximl connected expnsion of E. 3. Wge's con~truction proceeds in two steps which re nlogous to those bove. The first of these tkes plce within the rtionls Q, where it is desired to find sets U hving the property *) If q E U then (-oo,q) n U ~ ~ ~ U n (q,oo). The similrity of this property to the concept of singulr set is cler. Theorem 4. There exists topology 0 for Q which refines the Eucliden topology E for Q nd which is mximl with respect to the property tht if U E 0 then U hs property (*). Now one cn inductively order the irrtionls {X : < 2 w } nd the clopen subsets {U : < 2 w } of 0 such tht for ech < 2 w, x E Cl (Q - U ) n Cl For ech U < 2 w, choose mximl filters A nd B such tht for A E A nd B E B, ) A ~ Q n (X,oo), B ~ Q n (-oo,x) b) A,B E 0 c) x E Cl A n Cl B d) A nd B re seprted by U Define topology p for R to hve bse cr U {A U {X } U B: w < 2, A E A, B E B }. Theorem 5. The topology p is mximl connected.

5 352 Guthrie, Stone, nd Wge The inductive ordering bove is nlogous to the ssocition of rels with Cntor sets in the construction of the connected nonsingulr topology of Theorem 3. The bsic similrity of the constructions seems cler. The construction of Wge is efficient, but seems to be tied to Q or other countble dense subset. By selecting the proper ultrfilter of dense subsets, the topology constructed from Theorems 3 nd 1 cn be mde to hve dispersion chrcter 2 w. 4. Now tht mximl connected topology for the rels hs been constructed, we cll ttentign to some corollries nd to some of the remining questions. If mximlly connected topology is put on ech line through the origin, nd the plne is given the wek topology with respect to these lines, the result is plinly mximlly connected. This generlizes esily to the following observtion. Theopem 6. Let S be stplike subset of ny Eucliden spce. Then S dmits mximlly connected expnsion. We might lso sk how much more thn Husdorff seprtion cn be obtined. Since our mximl connected topology is finer thn, we utomticlly hve n exmple which is Urysohn nd functionlly Husdorff. Question 1. Does there exist (semi-) pegulr Husdorff mximlly connected spce? finer thn the rel line? Both of these constructions re confined to the rels, nd key step in ech cse is mde possible by the fct tht 2 w points re involved. Do mximl connected Husdorff

6 TOPOLOGY PROCEEDINGS Volume topologies exist for spces of other crdinlity? In prticulr, Question 2. Does there exist countble mximlly connected Husdorff spqce? References 1. J. A. Guthrie, D. F. Reynolds, nd H. E. Stone, Connected expnsions of topologies, Bull. Austrl. Mth. Soc. 9 (1973), MR 48# J. A. Guthrie nd H. E. Stone, Spces whose connected expnsions preserve conneoted subsets, Fund. Mth. 80 (1973), MR 48 # P. C. Hmmer nd W. E. Singletry, Connectednessequivlent spoes on the line, Rend. Circolo Mt. Plermo (Series II) 17 (1968), J. P. Thoms, Mximl topologicl spces, Disserttion, University of South Crolin (1965). 5., Mximl oonneoted topologies, J. Austrl. Mth. Soc. 8 (1968), University of Texs t El Pso El Pso, TX University of Southwestern Louisin Lfyette, Louisin Institute for Medicine nd Mthemtics, Ohio University Athens, OH nd Yle University (current) New Hven, Connecticut 06520