Preservations of so-metrizable spaces

A space is called an so-metrizable space if it is a regular space with a ?-locally finite sequentially open network. This paper proves that so-metrizable spaces are preserved under perfect mappings and under closed sequence-covering mappings, which give an affirmative answer to a question on preservations of so-metrizable spaces under some closed mappings. Also, we prove that the closed image of an so-metrizable space is an so-metrizable space if it is a topological group.

Having one's cake and eating it: the paradox of contextualisation in socio-legal research

Traditional doctrinal scholarship provides an important service to practising lawyers by analysing legal rules and decisions, clarifying ambiguities within rules, structuring them in a logical and coherent manner and describing their interrelationship (Chynoweth, 2008). The systematisation and formulation of the law in terms of doctrine creates a conceptual basis for constructing a legal context that helps to determine which rules should be applied in a particular situation. In this sense, doctrinal studies emerge out of the study of legal texts (or black-letter law), which are generated by legislature, courts and other legal authorities, and feed back in to legal practice once they are used in deciding cases. The method of doctrinal research, being functional to legal practice, dominates academic law and legal education. Notwithstanding its role in supporting legal practice, the doctrinal approach is criticised for conveying a normatively closed image of law (Cotterrell, 1995, pp. 50–53), for constructing the legal context narrowly, for presenting the legal system as a body of rules which can be studied in isolation from the broader societal context of the legal system by the exegesis of authoritative texts (Bradney, 1998, p. 76; Vick, 2004), for ‘not being self-conscious about its assumptions’ (Twining, 1999, p. 44) and for cultivating what Geoffrey Samuel (2009) calls the ‘authority paradigm’. Internally, i.e. from the standpoint of the legal system and its functionaries, this paradigm (or legal context) is produced by way of self-reference and normative closure, continually reaffirming the authority of legal sources such as legal texts, previous legal decisions and/or legislation, and prioritising definitions and methods based on what William Twining called the ‘practical insider attitudes’ (Twining, 2000, p. 129; for a discussion, see Banakar, 2003, p. 8). Externally, i.e. from the standpoint of policy-makers and citizenry, it is upheld through the threat of violence against non-compliance, backed by the authority of the modern state. The authority paradigm's normative closure and its dependency on coercion encourage ‘rigidity and introspection rather than an open-minded attitude to academic methods and pursuits’ (Samuel, 2009, p. 432). It fosters an understanding of the law as a system, which exists independently of societal forces. In order to escape the intellectual constraints of the authority paradigm, many academic lawyers turn to social theory and social sciences, which in contrast to law are based on the ‘perspective of enquiry’ (Samuel, 2009; see also Banakar, 2009b). These scholars place the law in the broader sociocultural context of the legal system and study legal phenomena in relation to societal forces, which are the prerequisite for the existence of the legal system and the production of the narrow legal context of the law. However, such an evasive strategy often comes at a price.

Seminorms related to weak compactness and to Tauberian operators

Let E and F be Banach spaces. The semigroup Φ+(E,F) of semi-Fredholm operators consists of the bounded linear mappings E→F with closed image and finite-dimensional kernel. By a well known result of Yood we have that T∈Φ+(E,F) if and only if for any bounded set B⊂E the condition TB relatively compact implies that B is relatively compact. Lebow and Schechter[10] gave a quantitative version of the above qualitative characterization, namely the operator T belongs to Φ+(E,F) if and only if there is c ≥ 0 such thatfor all bounded B⊂E. Here γ is the well known Hausdorff measure of non-compactnesswith BE the closed unit ball of E.