The bicompletion of an asymmetric normed linear space

A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖^s) is a Banach space, where ‖x‖^s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖_Y). Furthermore the space (Y,‖·‖_Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.     

Bicompletion and Samuel Bicompactification

It is proved that the quasi-proximity space induced by the bicompletion of a quasi-uniform T ₀-space X is a subspace of the quasi-proximity space induced by the Samuel bicompactification of X. The result is then used to establish that the locally finite covering quasi-uniformity defined on the category Top ₀ of topological T ₀-spaces and continuous maps is not lower K-true (in the sense of Brümmer). It is also shown that a functorial quasi-uniformity F on Top ₀ is upper K-true if and only if FX is bicomplete whenever X is sober.     

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology of a PO-quasi-uniform space (X, , ≤), the bicompletion of (X, ) is also a PO-quasi-uniform space ( , ⪯) with a partial order ⪯ on that extends ≤ in a natural way.      

On paratopological vector spaces

We show that each first countable paratopological vector space X has a compatible translation invariant quasi-metric such that the open balls are convex whenever X is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a paratopological vector space X and prove that X is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally, paratopological vector spaces of finite dimension are considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Direct Reflections

A pointed endofunctor (and in particular a reflector) (R, r) in a category X is direct iff for each morphism f : X Y the pullback of R f against r _Y exists and the unique fill-in morphism u from X to the pullback is such that R u is an isomorphism. (This is close to the concept of a simple reflector introduced by Cassidy, Hébert and Kelly in 1985.) We give sufficient conditions for directness, and for directness to imply reflectivity. We also relate directness to perfect morphisms, and we give several examples and counterexamples in general topology.