A FUNDAMENTALLY TOPOLOGICAL PERSPECTIVE ON GRAPH THEORY by Antoine Vella A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 2005 c Antoine Vella 2005 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract We adopt a novel topological approach for graphs, in which edges are modelled as points as opposed to arcs. The model of “classical” topologized graphs translates graph isomor- phism into topological homeomorphism, so that all combinatorial concepts are expressible in purely topological language. This allows us to extrapolate concepts from finite graphs to infinite graphs equipped with a compatible topology, which, dropping the “classical” requirement, need not be unique. We bring standard concepts from general topology to bear upon questions of a combinatorial inspiration, in an infinite setting. We show how (possibly finite) graph-theoretic paths are, without any technical sub- terfuges, a subclass of a broad category of topological spaces, namely “paths”, that includes Hausdorff arcs, the real line and all connected orderable spaces (of arbitrary cardinality). We show that all paths, and the topological generalizations of cycles, are topologized graphs. We use feeble regularity to explore relationships between the topologies on the vertex set and the whole space. We employ compactness and weak normality to prove the existence of our analogues for minimal spanning sets and fundamental cycles. In this framework, we generalize theorems from finite graph theory to a broad class of topologi- cal structures, including the facts that fundamental cycles are a basis for the cycle space, and the orthogonality between bond spaces and cycle spaces. We show that this can be accomplished in a setup where the set of edges of a cycle has a loose relationship with the cycle itself. It turns out that, in our model, feeble regularity excludes several patholo- gies, including one identified previously by Diestel and K¨uhn, in a very different approach which addresses the same issues. Moreover, the spaces surgically constructed by the same authors are feebly regular and, if the original graph is 2-connected, compact. We consider an attractive relaxation of the T1 separation axiom, namely the S1 axiom, which leads to a topological universe parallel to the usual one in mainstream topology. We use local connectedness to unify graph-theoretic trees with the dendrites of continuum theory and a more general class of well behaved dendritic spaces, within the class of ferns. We generalize results of Whyburn and others concerning dendritic spaces to ferns, and show how cycles and ferns, in particular paths, are naturally S1 spaces, and hence may be viewed as topologized hypergraphs. We use topological separation properties with a distinct combinatorial flavour to unify the theory of cycles, paths and ferns. This we also do via a setup involving total orders, cyclic orders and partial orders. The results on partial orders are similar to results of Ward and Muenzenberger and Smithson in the more restrictive setting of Hausdorff dendritic spaces. Our approach is quite different and, we believe, lays the ground for an appropriate notion of completion which links Freudenthal ends of ferns simultaneously with the work of Polat for non-locally-finite graphs and the paper of Allen which recognizes the unique dendritic compactification of a rim-compact dendritic space as its Freudenthal compactification. iii Acknowledgements My research work during the course of my Ph.D. programme has benefited greatly from numerous long and detailed conversations with my supervisor, Bruce Richter. It is difficult to accurately assign credit for all the proofs which emerged from these conversations. Although the entire framework, the concepts introduced, and the general and specific questions addressed were all of my inspiration, on several occasions his contribution was crucial in accelerating the process of resolving a proof down to its technical details. I am deeply grateful for this contribution. Apart from this, I will be forever indebted to my supervisor for his faith in an unorthodox research project, his unfailing availability, unstinting support, amazing generosity and endless patience, and the simple pleasure of working together in the pursuit of beauty in mathematics. I would also like to express my gratitude to the New Zealand Institute for Mathematics and its Applications, in particular Paul Bonnington, co-director of the NZIMA Programme for Combinatorics, for hosting me at Auckland during the last months of 2004, during which I was compiling this document. I would also like to thank Amarpreet Rattan for his help with the thesis submission procedure during my absence from the University of Waterloo campus, and Giuditta Franco and Kathrin Genk for their consultation on the Greek and German languages, respectively. Finally, I would like to thank all three for helping me pull through the dreary years at Waterloo. iv I dedicate this work to my friend Lavina McClintock, whose tardiness one night several years ago, at the Irish Pub in Pisa, spawned the musings which eventually led to a thesis. v Contents 1 There and Back Again (from combinatorics to topology) 1 1.1 From Graph Theory to Topology: motivation and basic constructions . 1 1.1.1 A topologically biased philosophy . 4 1.1.2 Classical Topologies . 8 1.1.3 Some graph-theoretic concepts in the language of general topology . 18 1.2 Relaxing the compatibility requirement . ...... 20 1.3 Overview..................................... 23 2 Prepaths, Paths and Cycles 27 2.1 CutpointsandSeparations . 28 2.2 Orderability................................... 33 2.2.1 Prepaths................................. 40 2.2.2 Connected subsets, jumps, gaps and intervals . 43 2.2.3 Boundedprepaths ........................... 50 2.3 Paths....................................... 55 2.3.1 The Interval Topology and Local Connectedness . 55 2.3.2 Characterizing prepath compatible orders . ...... 56 2.3.3 Compactness .............................. 62 2.4 Cycles ...................................... 65 3 Edge Spaces and Separation Axioms 69 3.1 SeparationAxioms ............................... 72 3.1.1 Some pathological examples . 72 3.1.2 Hierarchical Separation Axioms . 81 3.2 FeeblyHausdorffSpaces ............................ 87 3.3 CompatibilityandEdgeSpaces . 89 3.3.1 Extending a topology from V to V ∪ E ................ 89 3.3.2 Edgespaces ............................... 91 3.4 Feeble Regularity and Quasiregularity . ...... 93 vi 3.4.1 Reconstructing the topology on V ∪ E................. 94 3.4.2 Analysing feeble regularity . 98 3.5 Pre-edges,clumpsandquotients . 100 3.6 FeeblyNormalSpaces ............................. 107 4 Compact Spaces 111 4.1 SeparatingClosedandCompactSets . 112 4.2 ExistenceTheorems............................... 113 4.2.1 TheKnaster-Kuratoswkigraph . 115 4.2.2 Weakly Hausdorff Spaces . 119 4.2.3 Existence in Feebly Regular Topologized Graphs . 121 4.3 CompactnessandQuotients . 126 4.4 TheDiestel-K¨uhnend-quotient . 138 4.4.1 SeparationAxioms . 140 4.4.2 Compactness .............................. 142 5 Cycle Spaces 149 5.1 Edgeycles,edgepathsandplants . 150 5.1.1 Edgecycles and cyclic orders . 150 5.1.2 Edgepaths and uv-edgepaths...................... 154 5.1.3 Plants and Fundamental Cyclesets . 157 5.2 CycleSpacesandBondSpaces. 160 5.2.1 Weak,AlgebraicandStrongSpans . 160 5.2.2 BondsandEdgecuts . 163 5.3 Orthogonality and Cycle Spaces . 168 5.3.1 WeakOrthogonality . 168 5.3.2 FundamentalBases . 171 5.3.3 Orthogonalpairs ............................ 173 5.4 Quotients and Cycle Spaces in Feebly Hausdorff Spaces . 176 5.5 Partialresults.................................. 182 5.5.1 Restrictions on V ............................ 182 5.5.2 StrongvsAlgebraicSpans . 186 6 Local connectedness and Ferns 189 6.1 Wherewestand................................. 189 6.1.1 Changing the topological universe . 189 6.1.2 Whatshouldtreesbe? . 190 6.1.3 Combinatorial Separation Axioms . 196 6.1.4 Introducingthenewcomers. 198 6.2 Ferns....................................... 199 vii 6.2.1 Separation Properties in Locally Connected Spaces . ....... 199 6.2.2 From S[a, b] to D[a, b] ......................... 203 6.2.3 D[a, b]inFerns ............................. 204 6.3 Topological Characterizations of Ferns . ....... 207 6.3.1 Bringing in S1 .............................. 209 6.4 Cutends ..................................... 213 6.5 A Partial Order Characterization of Ferns . 220 7 Directions for further research 227 7.1 TheJordanCurveTheorem .......................... 227 7.2 Menger’sTheorem ............................... 227 7.3 EndsandCompletions ............................. 228 7.4 GraphMinors.................................. 229 Appendix A: Binary relations 230 Appendix B: General Topology 232 Bibliography 235 Index 240 viii List of Figures 1.1 Two-way infinite path plus vertex at infinity . ...... 21 3.1 Theinfinitebondplusirregularvertex . ..... 73 3.2 Theovercrowdedfan .............................. 76 3.3 TheDiestel-K¨uhnobstruction.. 77 3.4 Thedoubled-uparc............................... 80 3.5 Theone-wayinfinitepathwithadoubleend . 81 3.6 A V -set inducing a quasiregular, non-feebly-regular clump-edge quotient.