Doctor of Philosophy (Ph.D.)

In this thesis, domain theory in the realm of T₀ spaces is studied. Our work is motivated mainly by recent theorems which ‘lift’ classical domain-theoretic results from the posetal setting to the realm of T₀ spaces. Some of such theorems were reported in a 2015 paper by Zhao and Ho, entitled ‘On topologies defined by irreducible sets’.

Our first main focus is the Zhao-Ho replacement principle which concerns the replacement of directed subsets of posets by irreducible subsets of T₀ spaces. In Chapter 4, this replacement is applied to yield the definitions of Irr-convergence class and Irr-continuous spaces. Herein, we unearth some relations between Irr- convergence class, Irr-continuous spaces, and two novel types of T₀ spaces, i.e., balanced spaces and nice spaces.

In Chapter 5, we revisit the aforementioned paper by Zhao and Ho, and specifically look into some gaps which were left unaddressed by them. We fill in the gaps by: (1) re-examining the Zhao-Ho replacement principle, and then intro- ducing SI-convergence class that induces the SI-topology, (2) defining the notion of SI∗-continuous spaces, and then crucially proving SI- and SI∗-continuities are one and the same notion.

In Chapter 6, we re-work the main results reported in both Zhao and Ho’s 2015 paper and those results reported in Chapter 5 in very much the same spirit as Erne’s work on s₂-continuous posets and s₂-convergence class in which he generalised continuous posets and Scott convergence class. To do this, we define the notions of SI₂-topology, δ-sobriety, SI₂-continuous spaces, and SI₂-convergence class, and then prove some results related to these four new notions.

As the Zhao-Ho replacement is put in the limelight of our exposition in Chapter 4, we were not particularly successful in obtaining a complete characterisation of spaces in which the Irr-convergence is topological – which is a problem. In Chapter 7, we intentionally move away from our obsession of applying just the Zhao-Ho replacement principle, and instead open up the horizon by working with more general convergence classes. With this intentional deviation, we manage to locate a complete characterisation of those spaces in which such a general convergence class is topological. We then apply this characterisation to the special case of Irr-convergence class, and solve the aforementioned problem. Also in the same chapter, we generalise the order-convergence class in posets and establish a characterisation of posets in which the generalised order-convergence class is topological. An immediate consequence of our new characterisation is the recovery of a recent result by Sun and Li, where they obtained some neces- sary and sufficient condition for posets in which the order-convergence class is topological.

In our last chapter, Chapter 8, we define some endofunctors on the category of T₀ spaces. Our work here is motivated by theory in posets. We tailor make a version of theory suited for T₀ spaces. Crucially, this theoretical development leads to some interesting KZ-monads and KZ-comonads, which in turn allow us to give characterisations of F-injective (resp., E-projective) T₀ spaces, where F (resp., E) is determined by a certain monad (resp., comonad).