Compactness in apartness spaces?

A major problem in the constructive theory of apartness spaces is that of finding a good notion of compactness. Such a notion should (i) reduce to ``complete plus totally bounded\u27\u27 for uniform spaces and (ii) classically be equivalent to the usual Heine-Borel-Lebesgue property for the apartness topology. The constructive counterpart of the smallest uniform structure compatible with a given apartness, while not constructively a uniform structure, offers a possible solution to the compactness-definition problem. That counterpart turns out to be interesting in its own right, and reveals some additional properties of an apartness that may have uses elsewhere in the theory

Constructive topology of bishop spaces

The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS. Instead of having a common space-structure on a set X and R, where R denotes the set of constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F), where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R. The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions, the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces. The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient, the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact, the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces. The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively the Urysohn extension theorem for Bishop spaces. The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included

Kleist in France

From Madame de Staël, Dumas, and Taine, to Giraudoux, Adamov, Ionesco, and the existentialists, this study provides a fascinating account of the progress of Kleist's reception in France from complete rejection in 1807 to spectacular critical and public acclaim in the 1950s. Richardson argues that Kleist's success in France disproves the traditional idea that his work is "unexportable" and shows the conditions that led to his positive reception outside Germany

Unsatisfied appeal of sense : the decadent image in the poetry and poetics of Oscar Wilde, Arthur Symons, and Ernest Dowson

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Kleist in France

From Madame de Staël, Dumas, and Taine, to Giraudoux, Adamov, Ionesco, and the existentialists, this study provides a fascinating account of the progress of Kleist's reception in France from complete rejection in 1807 to spectacular critical and public acclaim in the 1950s. Richardson argues that Kleist's success in France disproves the traditional idea that his work is "unexportable" and shows the conditions that led to his positive reception outside Germany

HISTORY URBANISM RESILIENCE VOLUME 03:

The 17th conference (2016, Delft) of the International Planning History Society (IPHS) and its proceedings place presentations from different continents and on varied topics side by side, providing insight into state-of-the art research in the field of planning history and offering a glimpse of new approaches, themes, papers and books to come. VOLUME 03: Change and Responsive Plannin

Impersonality and the extinction of self: A comparative analysis of the poetry of Alun Lewis and Keith Douglas

This thesis, comparative in method, examines a wide range of the poetry of Keith Douglas and Alun Lewis, and some of their prose writings. As Second World War poets, both sought a poetic register that voiced their testimony to changed realities, both internal and external. Degrees of commonality are traced between Douglas’s dominant impulse for ‘impersonality’ and Lewis’s increasing stylistic objectivity, alongside investigation of their shared underlying sense of loss, and of complicity as agents of war, even when their poetic voice is at its most impersonal. Diverse critical viewpoints are addressed, along with several psychoanalytical theories and relevant biographical commentary. Following an Introduction and Review of the Critical Field, each chapter is structured as a bipartite comparison, focusing first on Douglas, then on Lewis. Chapter 1 investigates Douglas’s impersonality as a controlled, ambivalently detached poetic register which, in its undertow and perceptual shifts, reveals the speaker’s submersed engagement and ethical complicity. Lewis’s poetry is seen to reveal a related impulse for increasingly subordinating the subjective voice in evocations of the painfully harsh realities he encountered. Chapter 2 explores the writers’ dialectical struggles to resolve or extinguish self-division, focusing particularly upon Douglas’s ‘bête noire’ and Lewis’s ‘enmity within’, configurations analysed as paradoxically creative/destructive ingredients of the poetic impulse. Chapter 3 then examines the poets’ epistemological and ontological preoccupations with death, ‘darkness’ and ‘being’, and their relevance to what is here termed ‘the extinction of Self’. Chapter 4 extends this enquiry to examine the poets’ representations of wartime separation and geographical dislocation as manifestations of ‘the exilic self’ and a mutual desire to extinguish internal crises. The conclusion drawn is that their shared, dual axis of poetic engagement and detachment reveals a deeply embedded, common impulse to voice and escape their burdens, both inherently personal, and as complicit agents of war

Myth and legend in post-war English poetry

The thesis investigates both how and why many of the best post-war poets have moved back to use myth and legend just when they seemed finally discredited. Chapter One briefly discusses the various scholarly theories of myth, and sharpens the critical terminology to be employed. It describes the personal myth-systems of Yeats, Graves and Lawrence and, conversely, the four systems most generally used by modern poets. The following four chapters study, in turn, four major approaches to the use of myth in poetry. Chapter 2 shows how Seamus Heaney' s work employs legend as archetype; the history of Ulster being erected as a timeless metaphor to illuminate the present Troubles. Chapter Three takes the poetry of Geoffrey Hill as an example of the development of newly created legend, culminating with that centred on King Offa of Mercia. Chapter Four examines Thom Gunn' s use of myth as archetype, showing how the timeless can be given contemporary force, either through existential philosophy or Californian psychedelia. Chapter Five explores Ted Hughes' creation of a new myth, a new reality, the crowning achievement. Chapter Six discusses why myth is still relevant, distinguishing its careful adoption into four modern stylistic traditions and its four major modes of relevance. Legend is seen as a form of place, myth as a form of time, and the best new poetry is recognised as utilizing both, a surprise invocation of the White Goddess