Download “Quasi-uniform spaces”

“Quasi-uniform spaces”
April 28, 2001
A quasi-uniformity on a set X is a filter U on X × X satisfying the conditions: Each U ∈ U is reflexive, and for each U ∈ U there is V ∈ U such
that V 2 := V ◦ V ⊆ U. Here V ◦ V := {(x, z) ∈ X × X : there is y ∈ X such
that (x, y) ∈ V and (y, z) ∈ V }. For each quasi-uniformity U the filter U −1
consisting of the inverse relations U −1 = {(y, x) ∈ X × X : (x, y) ∈ U } where
U ∈ U is called the conjugate quasi-uniformity of U. A quasi-uniformity U
is said to be a uniformity if U −1 = U (see [E, Chapter 8]). If U1 and U2 are
quasi-uniformities on X such that U1 ⊆ U2 , then U1 is called coarser than U2 .
The coarsest uniformity U s finer than U is generated by the subbase U ∪ U −1 .
A map f : (X, U) → (Y, V) between quasi-uniform spaces (X, U) and (Y, V) is
called quasi-uniformly continuous if for each V ∈ V there is U ∈ U such
that (f × f )U ⊆ V. Basic facts about the category Quu of quasi-uniform spaces
and quasi-uniformly continuous maps can be found in [3]. The theory of quasiuniform spaces represents a generalization of the theories of metric spaces and
partial orders, which allows one to unify common features of these theories like
fixed point theorems and completions. The survey article [4] provides a comprehensive exposition (without proofs) of the area. Omitting many interesting
topics like fuzzy quasi-uniformities, quasi-uniformities on frames and approach
quasi-uniformities and not discussing applications of quasi-uniform spaces outside topology we shall concentrate in the following on some major aspects of
the theory. Each quasi-uniformity U on a set X induces a topology τ (U) determined by the neighborhood filters U (x) = {U (x) : U ∈ U} (x ∈ X), where
U (x) = {y ∈ X : (x, y) ∈ U }. Each quasi-uniformly continuous map between
quasi-uniform spaces is continuous. A quasi-uniformity U is called small-set
symmetric if τ (U −1 ) ⊆ τ (U ); its conjugate is then called point-symmetric
[3, Definition 2.22]. J. Marı́n and S. Romaguera proved that every continuous
map from a Lebesgue [3, Definition 5.1] quasi-uniform space to a small-set
symmetric quasi-uniform space is quasi-uniformly continuous. Similarly, each
continuous open map from a compact uniform space to any quasi-uniform space
is quasi-uniformly open. Each topological space (X, τ ) is quasi-uniformizable,
since τ coincides with the topology induced by the Pervin quasi-uniformity
generated by the subbase {[(X \ G) × X] ∪ [X × G] : G ∈ τ }, and admits a
finest compatible quasi-uniformity called its fine quasi-uniformity. H. Künzi
proved that a topological space admits a unique quasi-uniformity if and only
if all its interior-preserving open collections are finite. The condition implies
hereditary compactness and is equivalent to this property in quasi-sober spaces.
nw(X)
A topological space X of network weight nw(X) admits at most 22
quasiuniformities. H. Künzi also characterized internally the topological spaces admitting a coarsest quasi-uniformity. His condition is implied by local compactness (i.e. each point has a neighborhood base consisting of compact sets)
and is equivalent to it in spaces in which the limit set of each convergent ultrafilter is the closure of some unique singleton (i.e., supersober spaces).
Let [0, ∞) denote the set of the nonnegative reals. A quasi-pseudometric
1
d on a set X is a function d : X × X → [0, ∞) that satisfies d(x, x) = 0 and
d(x, z) ≤ d(x, y)+d(y, z) whenever x, y, z ∈ X. Then d−1 defined by d−1 (x, y) =
d(y, x) whenever x, y ∈ X is the conjugate quasi-pseudometric of d. A quasipseudometric d on X is called a quasi-metric if x, y ∈ X and d(x, y) = 0 imply
x = y; it is called non-archimedean if d(x, z) ≤ max{d(x, y), d(y, z)} whenever
x, y, z ∈ X. Each quasi-pseudometric d on X generates a quasi-uniformity Ud
with base {{(x, y) ∈ X × X : d(x, y) < } :  > 0}. For each quasi-uniformity U
possessing a countable base there is a quasi-pseudometric d such that U = Ud .
Every quasi-uniformity can be represented as the supremum of a family of
quasi-pseudometric quasi-uniformities (see e.g. I. Reilly’s work cited in [4]).
Other approaches to quasi-uniformities use pair-covers, or concepts of quasipseudometrics taking values in structures more general than the reals (see e.g.
R. Kopperman’s continuity spaces). Many classical counterexamples in topology like the Sorgenfrey plane, the Niemytzki plane or the Michael line are
quasi-metrizable, i.e., their topology is equal to τ (Ud ) for some quasi-metric
d. A γ-space is a topological T1 -space admitting a local quasi-uniformity
[3, Definition 7.15] with a countable base. R. Fox and J. Kofner constructed γspaces that are not quasi-metrizable. A topological space is non-archimedeanly
quasi-pseudometrizable if and only if it possesses a σ-interior-preserving base; it
is quasi-pseudometrizable if and only if it admits a local quasi-uniformity with
a countable base so that the conjugate filter is also a local quasi-uniformity. No
completely satisfactory characterization of quasi-pseudometrizability in topological terms seems to be known. H. Junnila established that each developable
γ-space is quasi-metrizable. Several kinds of γ-spaces are known to be nonarchimedeanly quasi-metrizable, e.g., the suborderable γ-spaces and γ-spaces
having an orthobase. J. Kofner showed that a first-countable image of a quasimetrizable space under a continuous closed map is quasi-metrizable; hence perfect surjections preserve quasi-metrizability. He also proved that a Moore space
that is an image of a quasi-metrizable space under a continuous open map with
compact fibers need not be quasi-metrizable.
A quasi-uniform space (X, U) is called precompact (resp. totally bounded)
if for each U ∈ U , {U (x) : x ∈ X} has a finite subcover (resp. U s is precompact).
The concept of a totally bounded quasi-uniform space is (in the categorical
sense) equivalent to the notion of a quasi-proximity space (see [3]). For any
topological space, the Pervin quasi-uniformity is its finest compatible totally
bounded quasi-uniformity. Each quasi-uniformity U contains a finest totally
bounded quasi-uniformity Uω coarser than U, which induces the same topology
as U . If the quasi-proximity class {V is quasi-uniformity on X: Vω = Uω } of
a quasi-uniformity U on X possesses more than one member, then it contains
at least 2c quasi-uniformities. H. Künzi proved that a topological space (X, τ )
admits a unique totally bounded quasi-uniformity if and only if its topology τ
is the unique base of open sets for τ that is closed under finite unions and finite
intersections. (Here the convention ∩∅ = X is used.) The condition is strictly
weaker than hereditary compactness, even in T1 -spaces, but is equivalent to this
property in supersober compact spaces. For each nonzero cardinal κ there exists
a (T0 -)space admitting exactly κ totally bounded quasi-uniformities.
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A quasi-uniformity is called transitive if it has a base consisting of transitive
relations. Each topological space that does not admit a unique quasi-uniformity
admits at least 2c nontransitive and at least 2c transitive quasi-uniformities.
Each infinite Tychonoff space admits at least 2c transitive totally bounded and
at least 2c nontransitive totally bounded quasi-uniformities. Let (X, τ ) be a
topological space
open covers C
S and let A be a collection of interior-preserving
S
of X such that A is a subbase for τ. For any C set UC = x∈X ({x} × ∩{D :
x ∈ D ∈ C}). Then {UC : C ∈ A} is a subbase for a compatible transitive
quasi-uniformity UA on X (the so-called Fletcher construction). If A is
the collection of all finite (resp. locally finite; interior-preserving) open covers
of X, then UA is the Pervin (resp. locally finite; fine transitive) quasiuniformity of X (see [3]). Similarly, if A is the collection of all well-monotone
open covers, i.e., covers well-ordered by set-theoretic inclusion, then UA is the
well-monotone quasi-uniformity of X. Its conjugate is always hereditarily
precompact. A quasi-uniform space (X, U ) is totally bounded if and only if
it is doubly hereditarily precompact, i.e., both U and U −1 are hereditarily
precompact. The quasi-uniformity generated by a preorder ≤ is hereditarily
precompact if and only if ≤ is a well-quasi-order (i.e. ≤ has neither infinite antichains nor infinite strictly descending chains). A topological space
is called transitive if its fine quasi-uniformity is transitive. The property is
neither finitely productive nor hereditary. Countable unions of closed (resp.
open) transitive topological spaces are transitive. Transitivity is preserved under continuous closed surjections. Kofner’s plane is a nontransitive topological
space, since it is quasi-metrizable, but not non-archimedeanly quasi-metrizable.
Various topological spaces are known to be transitive (compare [4]): e.g., the
orthocompact semistratifiable; suborderable; hereditarily metacompact locally
compact regular ; and T1 -spaces having an orthobase.
A quasi-uniform space (X, U) is called bicomplete provided that U s is a
complete (see [E, Chapter 8.3]) uniformity. Each quasi-uniform T0 -space (X, U )
e Ue) in
has an (up to quasi-uniform isomorphism) unique bicompletion (X,
e Ue) is a bicomplete extension of (X, U ) in which
the sense that the space (X,
fs coincide. Furthermore
(X, U) is τ (Ues )-dense. The uniformities (Ue)s and U
if f : (X, U) → (Y, V) is a quasi-uniformly continuous map into a bicomplete
quasi-uniform T0 -space (Y, V), then there exists a (unique) quasi-uniformly cone → Y of f. Each quasi-pseudometric T0 -space possesses a
tinuous extension fe : X
quasi-pseudometric bicompletion. Much work was devoted to spaces admitting
bicomplete quasi-pseudometrics (quasi-uniformities). We mention only two results: A metrizable space admits a bicomplete quasi-metric if and only if it is an
absolute metric Fσδ ; the fine quasi-uniformity of each quasi-pseudometrizable
and each quasi-sober space is bicomplete. Bicompletions of totally bounded
T0 -quasi-uniformities U yield joincompactifications, i.e., τ (Ues ) is compact.
The first thorough investigations of this concept were conducted by Á. Császár
and S. Salbany. The Fell compactification of a locally compact (T0 -)space can
be constructed as the bicompletion of its coarsest compatible quasi-uniformity.
The locally compact supersober compact (also called skew compact) spaces
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provide an analogue in the category of T0 -spaces to the class of compact spaces
in the category of Hausdorff spaces. They are characterized as the T0 -spaces
that admit a totally bounded bicomplete quasi-uniformity; it is their coarsest
compatible quasi-uniformity.
Let T denote the (obvious) forgetful functor from the category Quu to the
category Top of topological spaces and continuous maps. A functorial quasiuniformity is a functor F : Top → Quu such that T F = 1, i.e., F is a
T -section. (All the quasi-uniformities constructed above by Fletcher’s method
satisfy this criterion.) G. Brümmer showed that the coarsest functorial quasiuniformity is the Pervin quasi-uniformity. Using the idea of initiality, he built
functorial quasi-uniformities by the spanning construction (see [1]). In the
following discussion about the bicompletion functor K, we restrict ourselves
to T0 -spaces. G. Brümmer called a T -section F lower (resp. upper) K-true
if KF X ≤ F T KF X (resp. KF X ≥ F T KF X) for any space X; sections that
satisfy both conditions are called K-true. (Here ≤ corresponds to the coarser
relation for quasi-uniformities.) If F is K-true, then T KF is a reflection. G.
Brümmer showed that a T -section F is lower K-true if and only if F is spanned
by some class of bicomplete quasi-uniform spaces and H. Künzi noted that F is
upper K-true if and only if F is finer than the well-monotone quasi-uniformity.
The fine (resp. Pervin; locally finite) quasi-uniformity is K-true (resp. lower
K-true, but not upper K-true; neither lower nor upper K-true); see [5].
The categories Top and Quu are related to two other categories, namely
the category of bitopological spaces (= bispaces) and bicontinuous maps and the
category of topological ordered spaces and continuous increasing maps. Many
concepts can be transferred (often with some difficulties) between these four
categories. We next give some examples: Each quasi-uniform space (X, U) induces the bispace (X, τ (U), τ (U −1 )). Exactly the (pairwise) completely regular
bispaces can be obtained in this way. L. Nachbin observed that for any quasiuniform T0 -space (X, U), the triple (X, T (U s ), ∩U ) determines a topological
T2 -ordered space (i.e., the partial order ∩U is τ (U s ) × τ (U s )-closed); his construction determines exactly the completely regularly (T2 -)ordered spaces. The
theory of the bicompletion allows one to build bicompactifications (resp. ordered
compactifications) in these two categories. A completely regularly ordered space
(X, T , ≤) is said to be strictly completely regularly ordered (in the sense
of J. Lawson) if the bispace (X, T ] , T [ ) is pairwise completely regular. Here T ]
resp. T [ denotes the upper resp. lower topology of X. H. Künzi and S. Watson
constructed completely regularly ordered spaces that are not strictly completely
regularly ordered; e.g., such spaces cannot be topological lattices. R. Fox characterized quasi-metrizable bispaces as the pairwise stratifiable doubly γ-bispaces,
but pointed out that there are pairwise stratifiable, pairwise developable bispaces
which are not quasi-metrizable.
The search continues for completions based on a less symmetric idea than
the bicompletion. Originally the interest was directed towards Pervin-Sieber
completeness. A filter F on a quasi-uniform space (X, U ) is called a P S-filter
if for each U ∈ U there is x ∈ X such that U (x) ∈ F. A quasi-uniform space
(X, U) is called P S-complete provided that each P S-filter has a cluster point.
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For locally symmetric [3, Definition 2.24] quasi-uniform spaces the property
is equivalent to convergence completeness, i.e., each P S-filter converges. P.
Fletcher and H. Künzi showed that the fine quasi-uniformity of any nontrivial
Tychonoff Σ-product is not P S-complete. To date no regular quasi-metrizable
space is known whose fine quasi-uniformity is not P S-complete. Each Tychonoff
sequentially P S-complete quasi-metric space is Čech complete. Moreover each
Tychonoff orthocompact Čech complete space with a Gδ -diagonal admits a convergence complete (non-archimedean) quasi-metric. Unfortunately, P S-filters
are often difficult to handle and no satisfactory theory of P S-completions exists. In recent investigations P S-completeness was often abandoned in favour of
S- or left K-completeness. The latter concepts discussed below allow involved
combinatorial arguments and with their help many classical results could be
generalized satisfactorily.
Any net (xd )d∈D in a quasi-uniform space (X, U) is called left K-Cauchy
(resp. right K-Cauchy) if for any U ∈ U there is d ∈ D such that d2 , d1 ∈ D
and d2 ≥ d1 ≥ d imply that (xd1 , xd2 ) ∈ U (resp. (xd2 , xd1 ) ∈ U ). A filter F on
a quasi-uniform space (X, U ) is said to be left (resp. right) K-Cauchy if for each
U ∈ U there is F ∈ F such that U (x) ∈ F (resp. U −1 (x) ∈ F) whenever x ∈ F .
A quasi-uniformity is called left (resp. right) K-complete provided that
each left (resp. right) K-Cauchy T
filter converges. A filter F on a quasi-uniform
space (X, U) is called stable if F ∈F U (F ) ∈ F whenever U ∈ U . We next
mention some observations (mainly due to H. Künzi and S. Romaguera) about
the introduced concepts that extend classical results, whose proofs however often
need nontrivial modifications: P S-completeness implies left K-completeness.
Any right K-Cauchy filter is stable; the converse holds for ultrafilters. A filter
on a quasi-uniform space (X, U) is both stable and left K-Cauchy if and only if
it is Cauchy in (X, U s ). Left (resp. right) K-completeness is equivalent to left
(resp. right) K-completeness defined with nets. For a quasi-pseudometric quasiuniform space (X, Ud ) left K-sequential completeness, left K-completeness and
the condition that each co-stable filter has a τ (Ud )-cluster point are equivalent.
Here a filter is called co-stable if it is stable in the conjugate space. A quasiuniform space (X, U) is hereditarily precompact if and only if each ultrafilter
is left K-Cauchy (equivalently, each filter is co-stable); τ (U) is compact if and
only if U is precompact and left K-complete. The property that each co-stable
filter clusters is preserved under quasi-uniformly open continuous surjections
between quasi-uniform spaces. This property implies left K-completeness and
for uniform spaces is equivalent to supercompleteness (i.e., completeness of
the Hausdorff uniformity). Each regular left K-complete quasi-pseudometric
space is a Baire space. A quasi-metrizable space admits a left K-complete quasimetric if and only if it possesses a λ-base in the sense of H. Wicke and J. Worrell
(see [5]). Although many classical results on completeness do not generalize to
right K-completeness, that property behaves better than left K-completeness
in function spaces and hyperspaces.
From the point of view of category theory besides the bicompletion at least
two further satisfactory asymmetric completions for quasi-uniform spaces are
known: The theory of the Smyth-completion of topological quasi-uniform
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spaces (in a quasi-pseudometric variant often called the Yoneda-completion)
and the theory of the Doitchinov-completion of quiet quasi-uniform spaces
(with its quasi-metric variant for balanced quasi-metrics). M. Smyth endowed a quasi-uniform space (X, U ) with a topology τ that is not necessarily its standard topology τ (U ), but is linked to U by some additional axioms. He defined the (now called) Smyth- or S-completion for a topological quasi-uniform T0 -space (X, U , τ ). P. Sünderhauf called a quasi-uniform
T0 -space equipped with its standard topology S-completable if its Smythcompletion also carries the standard topology; e.g. each weightable T0 -quasipseudometric in the sense of S. Matthews induces an S-completable quasiuniformity. For S-completable spaces the construction of the Smyth-completion
coincides with the bicompletion. A quasi-uniform T0 -space (X, U ) is S-completable
(S-complete) if and only if each left K-Cauchy filter is Cauchy in (X, U s ) (τ (U s )converges). Hence each S-complete quasi-uniform space is left K-complete
and bicomplete. The Smyth-completion can be interpreted as a kind of Kcompletion. Since convergent filters in a quasi-uniform space are not left KCauchy in general, a simple theory of left K-completion is unlikely to exist.
N. Ferrario and H. Künzi showed that if we equip an arbitrary T0 -space with
the well-monotone quasi-uniformity (and its standard topology), then we obtain the sobrification equipped with its well-monotone quasi-uniformity as its
Smyth-completion (= bicompletion). Extending this result, R. Flagg, R. Kopperman and P. Sünderhauf introduced the concept of a well-monotone quasiuniformity Wτ for any topological quasi-uniform T0 -space and proved that its
Smyth-completion can be obtained via the bicompletion of Wτ .
A kind of bitopological extension of the bicompletion for a subclass of quasiuniform spaces was found by D. Doitchinov. Call a filter G on a quasi-uniform
space (X, U ) a D-Cauchy filter if there exists a co-filter F on X (i.e., for
each U ∈ U there are F ∈ F and G ∈ G such that F × G ⊆ U ). Then (F, G)
is said to be a Cauchy filter pair on (X, U). The quasi-uniformity U is called
D-complete if each D-Cauchy filter converges in (X, τ (U )). Of course, each convergence complete quasi-uniform space is D-complete. A quasi-uniform space
(X, U) is called quiet if for each U ∈ U there is V ∈ U such that for each
Cauchy filter pair (F, G) on (X, U) and all x, y ∈ X with V −1 (y) ∈ F and
V (x) ∈ G we have (x, y) ∈ U. Each quiet quasi-uniform T0 -space (X, U) has
a standard D-completion, now called its Doitchinov-completion, i.e., there is
an (up to quasi-uniform isomorphism) uniquely determined quiet D-complete
quasi-uniform T0 -space (X + , U + ) in which X is doubly dense; X + has the reflection property for quasi-uniformly continuous maps into quiet D-complete
quasi-uniform T0 -spaces. The Doitchinov-completion of the conjugate of a quiet
quasi-uniform T0 -space can be identified with the conjugate of its Doitchinovcompletion. Unfortunately quietness is a rather restrictive property, since e.g.
each quiet totally bounded quasi-uniform space is uniform by an observation due
to P. Fletcher and W. Hunsaker. Extension theories for (more) general quasiuniformities (and quasi-pseudometrics) were mainly developed in Hungary, e.g.
by J. Deák (see [2]). D. Doitchinov himself also investigated a D-complete extension for stable (i.e. each D-Cauchy filter is stable) quasi-uniform T0 -spaces.
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For quasi-pseudometric spaces stability is a strong property; e.g. each regular
doubly stable quasi-pseudometric space is subparacompact and closed sets are
Gδ -sets.
The completeness properties discussed above were studied extensively for
functorial quasi-uniformities; e.g., the well-monotone quasi-uniformity of any
topological space X is left K-complete; it is S-complete if and only if X is
quasi-sober (see [5]). The fine quasi-uniformity need neither be right K- nor
D-complete. We next deal with two further interesting classes of quasi-uniform
spaces investigated in recent years. A quasi-uniformity U on a set X is called
uniformly regular if for each U ∈ U there is V ∈ U such that clτ (U) V (x) ⊆
U (x) whenever x ∈ X. Quiet quasi-uniform spaces are doubly uniformly regular. For a topological space the property of admitting a uniformly regular
quasi-uniformity lies strictly between regularity and complete regularity. Each
uniformly regular quasi-uniformity is bicomplete if D-complete, and S-complete
if left K-complete. P. Fletcher and W. Hunsaker established that the fine transitive quasi-uniformity of a topological space is quiet and convergence complete
if it is uniformly regular. The fine quasi-uniformity of each regular preorthocompact semistratifiable space and each regular T1 -space with an orthobase is uniformly regular (see [5]). A quasi-uniformity U is called monotonic if there
exists an operator M : U → U such that M (U ) ⊆ M (V ) whenever U, V ∈ U
and U ⊆ V, and such that M (U )2 ⊆ U whenever U ∈ U. H. Junnila and H.
Künzi established that each T1 -space with an orthobase admits a monotonic
quasi-uniformity.
Only few results about uniformities on function spaces and hyperspaces generalize without complications to quasi-uniformities, as e.g. work of J. Cao,
S. Naimpally, B. Papadopoulos and H. Render showed. The following results may serve as examples: For a topological space X and a quasi-uniform
space (Y, V) the quasi-uniformity of quasi-uniform convergence on the
set Y X of all functions from X to Y is right K-complete if and only if V is right
K-complete; the corresponding result does not hold for left K-completeness.
The subspace C(X, Y ) of continuous maps from X to Y is closed in Y X if
V is locally symmetric, but not in general. The quasi-uniformity of quasiuniform convergence on compacta induces on C(X, Y ) the compact-open
topology if V is small-set symmetric, but not for arbitrary V. Let (X, U) be a
quasi-uniform space. On the set P0 (X) of nonempty subsets of X the Hausdorff quasi-uniformity UH is generated by the base {UH : U ∈ U} where
UH = {(A, B) : B ⊆ U (A) and A ⊆ U −1 (B)}. The Hausdorff quasi-uniformity
is precompact (resp. totally bounded; joincompact) if and only if the underlying
space has the corresponding property. Bicompleteness and left K-completeness
(even for quasi-pseudometric spaces) behave rather badly under the Hausdorff
hyperspace construction. Generalizing a well-known result due to B. Burdick
and J. Isbell on supercompleteness of uniform spaces, H. Künzi and C. Ryser established that UH is right K-complete if and only if each stable filter on (X, U )
clusters. H. Künzi and S. Romaguera characterized those quasi-uniformities
U for which τ (UH ) is compact (resp. UH is hereditarily precompact); both
results rely on combinatorial facts about well-quasi-ordering. Studies were con7
ducted about those quasi-uniformities for which UH induces on the set K0 (X)
of nonempty τ (U)-compact sets the Vietoris topology and about quasi-uniform
variants of K. Morita’s result that (K0 (X), UH |K0 (X)) is complete for any complete uniform space (X, U) (see [5]). Furthermore authors studied the multifunction space (equipped with its natural Hausdorff quasi-uniformity) and considered asymmetric versions of the result that each almost uniformly open (multivalued) map with closed graph from a supercomplete uniform space into an
arbitrary uniform space is uniformly open.
Quasi-uniformities were also investigated in topological algebraic structures.
The study of paratopological groups and asymmetric normed (real) vector spaces
and cones with the help of canonical quasi-uniformities was particularly fruitful. A paratopological group is a group equipped with a topology such
that the group multiplication is continuous; an asymmetric norm k · k possibly satisfies kaxk = akxk only for nonnegative real a. We shall concentrate
here on paratopological groups, but note in passing that asymmetric norms
provide e.g. an efficient approach to the complexity spaces introduced by M.
Schellekens. Among other things, J. Marı́n and S. Romaguera showed that the
ground set of the bicompletion of the two-sided quasi-uniformity of a paratopological (T0 -)group naturally carries the structure of a paratopological group; the
quasi-uniformity of the bicompletion yields its two-sided quasi-uniformity. Furthermore they observed that each first-countable paratopological group admits
a left-invariant quasi-pseudometric compatible with its left quasi-uniformity.
Together with H. Künzi they generalized parts of their theory about paratopological groups to topological semigroups (with neutral element). H. Künzi, S.
Romaguera and O. Sipacheva noted that the two-sided quasi-uniformity of a
regular paratopological group G is quiet. They established that its Doitchinovcompletion can be considered a paratopological group if G is abelian; in general
however the multiplication does not extend to the completion. S. Romaguera,
M. Sanchis and M. Tkachenko proved that for any topological (resp. quasiuniform) space X the free and the free abelian paratopological group over X
exist (see [5]). H. Künzi continued work of P. Fletcher and K. Porter by showing
that the group of all quasi-uniform isomorphisms of a bicomplete quasi-uniform
(T0 -)space equipped with the quasi-uniformity of quasi-uniform convergence
yields a paratopological group whose two-sided quasi-uniformity is bicomplete.
References
[1] G.C.L. Brümmer, Categorical aspects of the theory of quasi-uniform
spaces, Rend. Ist. Mat. Univ. Trieste 30 Suppl. (1999), 45–74.
[2] Á. Császár, Old and new results on quasi-uniform extension, Rend. Ist.
Mat. Univ. Trieste 30 Suppl. (1999), 75–85.
[3] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Lecture Notes
Pure Appl. Math. 77, Dekker, New York, 1982.
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[4] H.-P.A. Künzi, Nonsymmetric distances and their associated topologies:
about the origins of basic ideas in the area of asymmetric topology, in
Handbook of the History of General Topology, Vol. 3, eds. C. Aull and
R. Lowen, Kluwer, Dordrecht (to appear).
[5] H.-P.A. Künzi, Quasi-uniform spaces in the year 2001, preprint.
Hans-Peter A. Künzi
University of Cape Town, SA
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