Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: While the notions of open and closed subsets in a topological space are dual to each other, they take on another meaning when points and complements are no longer available. Closure operators have been extensively used to study topological notions on categories. Though this has recovered a fair amount of topological results and has brought an economy of e ort and insight into Topology, it is thought that certain properties, such as convergence, are naturally associated with neighbourhoods. On the other hand, it is interesting enough to investigate certain notions, such as that of closed maps, which in turn are naturally associated with closure by means of neighbourhoods. We propose in this thesis a set of axioms for neighbourhoods and test them with the properties of connectedness and compactness. / AFRIKAANSE OPSOMMING: Al is die twee konsepte van oop en geslote subversamelings in 'n topologiese ruimte teenoorgesteldes van mekaar, verander hul betekenis wanneer punte en komplemente nie meer ter sprake is nie. Die gebruik van afsluitingsoperatore is alreeds omvattend in die studie van topologiese konsepte in kategorieë, toegepas. Alhoewel 'n redelike aantal topologiese resultate, groeiende belangstelling en groter insig tot Topologie die gevolg was, word daar geglo dat seker eienskappe, soos konvergensie, op 'n natuurlike wyse aan omgewings verwant is. Nietemin is dit van belang om sekere eienskappe, soos geslote afbeeldings, wat natuurlik verwant is aan afsluiting, te bestudeer. In hierdie proefskrif stel ons 'n aantal aksiomas oor omgewings voor en toets dit gevolglik met die eienskappe van samehangendheid en kompaktheid.

Topological insulators and superconductors, which are featured by not only the topological characteristics of their gapped bulk band structure but also the special edge or surface states, have attracted great attention in the past few years. A complete classification of topological insulators and superconductors in terms of symmetry and spatial dimension has been established, while the application of their surface states remains a challenge. The gapless phases which have topologically stable Fermi surfaces could also exhibit peculiar surface states and topological transport phenomena in the bulk. In this thesis, the topological DIII-classs superconducting chains and the application of its Majorana edge states are studied. On the other hand, Weyl semimetals, as the representative example of topological gapless phases, and its exotic transport phenomena are also investigated. Majorana edge states have been a focus of condensed matter research for their potential applications in topological quantum computation, which appear in the topological DIII-class superconducting chains protected by both the particle-hole and time reversal symmetries. We utilize two charge-qubit arrays to explicitly simulate one type of DIII-class superconducting chains and the universal quantum operations performed on the Majorana edge states. It is shown that combined with one braiding operation, universal single-qubit operations on a Majorana-based qubit can be implemented by a controllable inductive coupling between two charge qubits at the ends of the arrays. It is further shown that in a similar way, a controlled-NOT gate for two topological qubits can be simulated in four charge-qubit arrays. Although the current scheme may not truly realize topological quantum operations, we elaborate that the operations in charge-qubit arrays are indeed robust against certain local perturbations. Weyl semimetals possess nontrivial Fermi surface topology in that the pair of Weyl points with opposite topological charges is separated from each other in momentum space. The physical manifestations of this Fermi surface topology are protected surface states and exotic transport phenomena including the anomalous Hall effect as well as the chiral magnetic effect. By studying the path integral measure under the chiral transformation, it is shown that these transport phenomena can be described by the chiral anomaly which appears when the chiral Weyl fermion couples to the topologically nontrivial gauge field. The case of the gauge anomaly for the Weyl fermion coupled to a non-Abelian gauge field is also discussed. / published_or_final_version / Physics / Doctoral / Doctor of Philosophy

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