Convexity and quasi-uniformizability of closed preordered spaces
In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as quasi-uniformizability, allows one to compactify the topological space and to extend its order dynamics. In this work we study locally compact -compact spaces endowed with a closed preorder. They are known to be normally preordered, and it is proved here that if they are locally convex, then they are convex, in the sense that the upper and lower topologies generate the topology. As a consequence, under local convexity they are quasi-uniformizable. The problem of establishing local convexity under antisymmetry is studied. It is proved that local convexity holds provided the convex hull of any compact set is compact. Furthermore, it is proved that local convexity holds whenever the preorder is compactly generated, a case which includes most examples of interest, including preorders determined by cone structures over differentiable manifolds. The work ends with some results on the problem of quasi-pseudo-metrizability. As an application, it is shown that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable.
Topological preordered spaces are ubiquitous. They appear in the study of dynamical systems (1), general relativity (2), microeconomics (3); (4), thermodynamics (5) and computer science (6). In these applications it is important to establish if a topological preordered space is quasi-uniformizable, namely, if there is a quasi-uniformity such that and . Taking into account a characterization of quasi-uniformizability established by Nachbin (7), this problem is equivalent to that of establishing if the topology and the preorder of the space are determined by the family of continuous isotone functions.
Hausdorff quasi-uniformizable spaces are compactifiable (7); (8) and, in general, the possibility of restricting an analysis to the compact case brings several simplifications. In other circumstances, the boundary (remainder) involved in the compactification has special importance. For instance, a good definition of spacetime boundary in general relativity would allow us to identify the singular spacetime points (9).
Quasi-uniformizable spaces are -preordered spaces, thus must be closed in order to have any chance to come from a quasi-uniformity. In the various fields that in one way or the other are connected with topological preordered spaces, it has been discovered that it is indeed very convenient to study some new closed preorder related to the original preorder. This is the strategy of ‘prolongations’ introduced by Auslander in dynamical systems (10), and rediscovered in a different setting in relativity theory, where Seifert (11); (12) introduced a closed relation related to the causal relation, and Sorkin and Woolgar (13) introduced the smallest closed relation containing the causal relation (see Sect. 1.1).
Given a -preordered space it is possible to infer preorder normality provided is a -space (14). We recall that a topological space is a -space if there is a (admissible) sequence of compact sets , , such that is open if and only if is open in . It is not restrictive to assume , and equal to any chosen compact set. In this work we shall use the fact that locally compact -compact spaces are -spaces. Indeed, under local compactness the properties: hemicompact -space, -space, -compact, and Lindelöf are equivalent (15). Since we do not assume that is Hausdorff, we remark that in our terminology a topological space is locally compact if each point has a compact neighborhood. It is strongly locally compact if at each point the neighborhood system of the point has a base made of compact neighborhoods (not necessarily closed).
Convex normally preordered spaces are quasi-uniformizable ((8), Prop. 4.7) (i.e. they are completely regularly preordered spaces (7)), and quasi-uniformizable spaces are convex closed preordered spaces. Unfortunately, although a -preordered -space is normally preordered, preorder normality does not imply quasi-uniformizability as convexity is missing. Indeed, we shall give an example of a -preordered locally compact -compact space which is not convex (see example 1.5).
This work is devoted to the proof of the convexity and hence quasi-uniformizability of a large class of locally compact -compact closed preordered spaces.
The main result of this work is the proof that for these preordered spaces local convexity and convexity are equivalent (Cor. 2.14). We then proceed to study local convexity, showing that it follows from antisymmetry plus some other assumptions. We prove that local convexity holds for -preserving spaces (Theor. 3.3), namely for those spaces for which the convex hull of any compact set is compact. The definition of -preserving space is quite important for the connection with global hyperbolicity in relativity theory (16) (see Sect. 1.1).
Furthermore, we show that if the order is compactly generated then local convexity holds (Cor. 4.12). Joining this result with the previous one we infer that if, roughly speaking, both the topology and order are generated ‘locally’ then convexity holds (Cor. 4.14). This case includes most examples of topological preordered spaces of interest, including those in which the preorder is induced by a distribution of tangent cones on a differentiable manifold (17). We shall compare our findings with similar results obtained by Akin and Auslander in the study of dynamical systems (18).
Finally, under second countability we obtain a result on the quasi-pseudo-metrizability of the space which generalizes Urysohn’s theorem (Theor. 5.1), and under the -space condition we are able to assure the strict quasi-pseudo-metrizability of the space (Theor. 5.3). As an application, we prove that globally hyperbolic spacetimes (see Sect. 1.1 for the definition) are strictly quasi-pseudo-metrizable.
1.1 Some reference results on mathematical relativity and causality theory
At places we shall illustrate our findings using the topological ordered space given by the spacetime manifold ordered with a causal relation. Therefore, it is worth recalling some definitions and result from this field. The reader can skip this section on first reading, returning to it whenever this application is mentioned.
Let be a Hausdorff, connected, paracompact (, ) manifold and let be a (, ) Lorentzian metric, namely a pseudo-Riemannian metric of signature . Non vanishing tangent vectors split into spacelike, lightlike or timelike depending on the sign of , , respectively positive, null or negative. Lightlike or timelike vectors are called causal. Assume that a continuous timelike vector field can be defined over , and call future the cone of causal vectors including it. If this is not possible there is always a double covering of with this property, thus this is not a severe restriction. Once such a choice of future cone has been made, the Lorentzian manifold is time oriented. A spacetime is a time oriented Lorentzian manifold. The simplest example of spacetime is the 2-dimensional Minkowski spacetime, namely with coordinates , metric and time orientation given by the global timelike vector .
Let us observe that once a time orientation is given, any causal vector is either future directed or past directed depending on whether it belongs to the future cone. This terminology extends to curves depending on the character of their tangent vector, provided it is consistent throughout the curve.
The causal relation over is defined through: if there is a future directed causal curve from to or . The chronology relation over is defined through: if there is a future directed timelike curve from to . We have , and is open in the product topology (19); (16). Unfortunately, the causal relation is not necessarily closed, as can be easily realized considering the spacetime which is obtained removing a point from the 2-dimensional Minkowski spacetime.
The relation is by definition (13) the smallest closed and transitive relation containing and it exists because provides an example of closed and transitive relation containing . Unfortunately, it is difficult to work with since it is defined through its closure and transitivity properties rather than through the more intuitive notion of causal curve. Seifert (11) found another route to build a closed and transitive relation. Let us write if the timelike cones of contain the causal cones of , and let be the causal relation for . Seifert proved that is indeed closed, transitive and contains .
A spacetime is said to be causal if it does not contain any closed causal curve. It is stably causal if there is such that is causal, namely if it is possible to open the light cones everywhere over without introducing closed causal curves. A relation is antisymmetric if and implies . It can be proved that the spacetime is causal (resp. stably causal) iff (resp. ) is antisymmetric (12). It also turns out (20) that stable causality holds iff is antisymmetric, and in this case . Thus is really the most natural closed and transitive relation that can be introduced in a stably causal spacetime.
Let us write and , and if , let . A spacetime is causally continuous if the relation
is antisymmetric (a property known as weak distinction) and coincides with (a property known as reflectivity). It is not hard to prove (21) that is transitive, thus under causal continuity is closed, transitive and contains . As a consequence, it is the smallest relation with such properties, , and hence causal continuity implies stable causality.
A spacetime is causally simple if it is causal and is closed. Clearly, under causal simplicity , thus causal simplicity implies causal continuity (note that under causal simplicity we have also ).
Another important causality property is global hyperbolicity. A spacetime is globally hyperbolic if it is causal and for every compact set , its convex causal hull is compact. It can be shown that every globally hyperbolic spacetime is causally simple (19). These spacetimes are the most studied in mathematical relativity because a spacetime is globally hyperbolic iff it admits a Cauchy hypersurface, namely a topological hypersurface intersected by any inextendible (i.e. with no endpoint) causal curve in exactly one point (16). Therefore, they are the spacetimes for which the Cauchy problem of general relativity and that of wave equations makes sense.
1.2 Preliminaries on topological preordered spaces
A topological preordered space is a triple where is a topological space and is a preorder on , namely a reflexive and transitive relation. A preorder is an order if it is antisymmetric (that is, and ). For a topological preordered space our terminology follows Nachbin (7). With and we denote the increasing and decreasing hulls, and we define . The topological preordered space is -preordered (or semiclosed preordered) if and are closed for every , and it is -preordered (or closed preordered) if the graph of the preorder is closed.
Let , we define and analogously for . A subset , is called increasing if and decreasing if . It is called monotone if it is increasing or decreasing. With we denote the smallest closed increasing set containing , and with we denote the smallest closed decreasing set containing . A subset is convex if it is the intersection of a decreasing and an increasing set in which case . A subset is a -set (22) if it is the intersection of a closed decreasing and a closed increasing set in which case . The neighborhood of a point which is a -set is a -neighborhood, and a -set which is compact is a -compact set. In the notation of this work the set inclusion , is reflexive, i.e. .
A topological preordered space is a normally preordered space if it is -preordered and for every closed decreasing set and closed increasing set which are disjoint, , it is possible to find an open decreasing set and an open increasing set which separate them, namely , , and .
Given a reflexive relation on , a function such that is an isotone function. An isotone function such that is a utility function.
In a normally preordered space, closed disjoint monotone sets as and above can be separated by a continuous isotone function , that is , (this is the preorder analog of Urysohn’s separation lemma, see ((7), Theor. 1)). Normally preordered spaces are -preordered spaces, and -preordered spaces are -preordered spaces.
A topological preordered space is convex at , if for every open neighborhood , there are an open decreasing set and an open increasing set such that (this definition is due to Nachbin (23) and is used in (24); (22); (25), though the terminology is not uniform in the literature). It is locally convex at if the set of convex neighborhoods of is a base for the neighborhoods system of this point (23); (7). It is weakly convex at if the set of convex open neighborhoods of is a base for the neighborhoods system of this point (23); (26). The topological preordered space is convex (locally convex, weakly convex) if it is convex (resp. locally convex, weakly convex) at every point. Clearly, convexity (at a point) implies weak convexity (at a point) which in turn implies local convexity (at a point). Notice that according to this terminology the statement “the topological preordered space is convex” differs from the statement “the subset is convex” (which is always true).
A quasi-uniformity (7); (8) is a pair such that is a filter on , whose elements contain the diagonal , and such that if then there is , such that . A quasi-uniformity is a uniformity if implies , where . To any quasi-uniformity corresponds a dual quasi-uniformity .
From a quasi-uniformity it is possible to construct a topology in such a way that a base for the filter of neighborhoods at is given by the sets of the form where with . In other words, if for every there is such that .
Given a quasi-uniformity , the family given by the sets of the form , , is the coarsest uniformity containing . The symmetric topology of the quasi-uniformity is . Moreover, the intersection is the graph of a preorder on (see (7)), thus given a quasi-uniformity one naturally obtains a topological preordered space . The topology is Hausdorff if and only if the preorder is an order (7).
Nachbin proves ((7), Prop. 8) that a topological preordered space comes from a quasi-uniformity , in the sense that and , if and only if is a completely regularly preordered space (-preordered space, Tychonoff-preordered space), namely if and only if the following two conditions hold:
coincides with the initial topology generated by the set of continuous isotone functions ,
if and only if for every continuous isotone function , .
Completely regularly preordered spaces are convex -preordered spaces (convexity follows from (i) see ((7), Prop. 6, Cap.II), and the closure of the preorder follows from (ii)). Contrary to what happens in the usual discrete-preorder case, normally preordered spaces need not be completely regularly preordered spaces (see example 1.5), nevertheless the preorder analog of Urysohn’s separation lemma implies that convex normally preordered spaces are completely regularly preordered spaces. Completely regularly ordered spaces admit the Nachbin’s -ordered compactification (see (8) and (27) for the preorder case).
1.3 Preliminary results on convexity
A theorem by Nachbin states that every compact -ordered space is convex ((7), p. 48). Unfortunately, this theorem assumes the compactness of the space from the start, and hence it is not really useful in applications. There one would like to pass through convexity exactly to prove quasi-uniformizability, so as to introduce and work in the compactified space.
The most common strategy is then that of adding some additional conditions to the preorder such as the -space and -space conditions (28) (compare with the definitions of continuous and anti-continuous preorder in (26); (24)). A topological preordered space is a -space (-space) if for every closed (open) subset , and are closed (resp. open).
The following theorem and proof are due to H.-P. Künzi ((29), Lemma 2). They are included for the reader convenience.
Every normal -ordered -space is convex.
Let be an open neighborhood of . The closed sets and are disjoint. By normality these sets can be separated by open sets, say and , then and . The set is closed and is disjoint from . By the -space assumption is closed and is disjoint from thus is an open decreasing set such that . Analogously there is an open increasing set such that . Thus . Hence the space is convex. ∎
Unfortunately the -space condition is too strong as not even with the product order is a -space (consider the increasing hull of the closed set ).
Concerning the -space property we have the following simplification.
Every locally convex -space is convex.
Let and let be an open neighborhood of . By local convexity there are a convex set and an open set such that . Since is an -space the sets and are respectively open increasing and open decreasing. Furthermore, , which proves that is convex. ∎
In this connection, the next interesting result due to Burgess and Fitzpatrick ((24), Cor. 4.4) is worth mentioning
Every locally compact convex -ordered -space is completely regularly ordered.
The -space property is sometimes justified in applications. For
instance, in general relativity (see Sect. 1.1) the closure
of the causal relation in a causally continuous
spacetime provides a preorder which turns spacetime into a
topological closed preordered -space.
In this work we shall try to avoid as much as possible the simplifying -space and -space assumptions, and we shall instead impose weak conditions on the preorder and the topology in order to attain convexity. We shall meet again the I-space assumption at the end of this work, where it is used in connection with strict quasi-pseudo-metrizability.
We end the section with examples which show that a normally preordered space need not be convex. An example can be found in ((8), Example 4.9).
A locally compact -compact -ordered space which is not locally convex can be found in ((18), p. 59). The next example is particularly interesting because the topology has nice properties.
Let with the induced topology which we denote . The topology is particulary well behaved, it is connected, metrizable, locally compact, -compact, second countable. Define on the order through the following increasing hulls
With this definition the decreasing hulls are
It is easy to check that is reflexive, transitive and antisymmetric and hence an order. with this order is a -ordered space, indeed let with . If then necessarily as , . If then for sufficiently large , thus and then that is . If then we can assume, up to a subsequence, that either for all , (and hence ), or . In the former case and then thus , while in the latter case passing to the limit the equation we get that is which concludes the proof. Let us observe that is second countable and locally compact which implies that is a normally ordered space (14). Nevertheless, convexity does not hold at and in fact even local convexity fails there because every convex neighborhood of contains points ‘arbitrarily close to the lower edge at 0’.
2 From local convexity to convexity
The mentioned examples of -preordered locally compact -compact spaces which are not convex are also non-locally convex. This fact suggests that, perhaps, we could obtain convexity by assuming local convexity plus some topological property. This is indeed the case and in this section we shall prove that a locally convex -preordered locally compact -compact space is necessarily convex. This result is important because it is often much easier to prove local convexity than convexity. The next two sections will then show how to obtain local convexity for a large class of topological preordered spaces.
We need to state the next two propositions which generalize to preorders two corresponding propositions due to Nachbin ((7), Prop. 4,5, Chap. I). Actually the proofs given by Nachbin for the order case work unaltered. For this reason they are omitted.
Let be a -preordered space. For every compact set , we have and , that is, the decreasing and increasing hulls are closed.
Let be a -preordered compact space. Let where is increasing and is open, then there is an open increasing set such that . An analogous statement holds in the decreasing case.
We start with a convex analog to the previous proposition.
Let be a normally preordered space, let be a closed decreasing set and let be a closed increasing set. Finally, let be a compact set and let be an open set such that , then there are an open decreasing set and an open increasing set , such that .
The set being a closed subset of a compact set is compact. Let , we know that or . In the former case there is an open increasing set and an open decreasing set such that . If (and hence ) there is an open decreasing set and an open increasing set such that . Since is compact there are some , , , such that the sets cover . The index set splits into the disjoint union of the two subsets , , where iff . Let us define and . The subsets are such that and . Let us prove that . Indeed, suppose , then is contained in some , , that does not intersect or depending on whether or not, thus which implies . By applying preorder normality we find open decreasing set such that and open increasing set such that , thus . ∎
Let be a -preordered compact space, let be a closed decreasing set and let be a closed increasing set. Finally, let be an open set such that . Then there are an open decreasing set and an open increasing set , such that .
It is well known that under Hausdorffness local compactness and strong local compactness are equivalent. Every -ordered space is Hausdorff thus under antisymmetry these notions of local compactness coincide. We can actually prove that this equivalence holds at a single point.
Let be a subspace of . In the next theorems with “on ” we shall mean “with respect to regarded as a subspace, namely with its induced topology and induced preorder”. On the increasing hull of a subset will be denoted and analogously for the decreasing hull, , and for the corresponding closure versions, and .
Let be a -preordered space. If admits a compact neighborhood then for every open set there is a compact neighborhood of contained in . In particular, under antisymmetry at , local compactness at implies strong local compactness at .
Let be a compact neighborhood of , . Let be an open set such that and define . Let , . Since , we have which implies . We work on the -preordered compact space and apply lemma 2.4. There are an open decreasing set (on ) and an open increasing set (on ), such that . Since , the -open set is actually open in . The set being a closed subset of is compact, and containing , it is actually a compact neighborhood of contained in and hence . ∎
Let be a -preordered space, a compact subset, an open set on , a convex set on , and a point in , such that where . Then , and there are an open convex neighborhood (on ) of contained in , and a -compact neighborhood (on ) of contained in .
If then , thus , that is .
Let be a (-)convex neighborhood of contained in , where convexity refers to the subspace . We are going to prove that is convex in . Indeed
thus if then which implies
by convexity of in , thus is indeed convex in .
Suppose that we prove the existence of an open convex neighborhood of contained in in the -preordered subspace . Since and is open in , is open in and also convex by the above argument.
Analogously, suppose that we prove the existence of a -compact neighborhood of contained in in the -preordered subspace with the additional property that it contains an open convex (in the subspace ) neighborhood of contained in . Since is compact on it is compact on . The equation proves that is closed in , and that it is a -compact set in (recall Prop. 2.1). Since it contains which contains it is a -compact neighborhood of .
Thus for the remainder of the proof we can work in the -preordered compact subspace . Let and , so that . Lemma 2.4 proves that there are an open convex neighborhood of contained in (according to the subspace ), and a -compact neighborhood of contained in (according to the subspace ), which finishes the proof. ∎
If local convexity holds at then is compact and every open neighborhood of is also an open neighborhood of .
Let be an open neighborhood of and let be a convex set such that , then , thus is also an open neighborhood for . Let us consider an open covering of , then there is some open set of the covering which includes and hence , thus every open covering admits a subcovering of only one element. ∎
Let be a -preordered space. If is locally compact and locally convex at , then the topology at admits a base of -compact neighborhoods, and a base of open convex neighborhoods (that is, weak convexity holds at ).
Let be any open neighborhood of . By local compactness there are a compact set and an open set such that . By local convexity there is a convex set and an open set such that . By local convexity (lemma 2.7) . By lemma 2.6 there are an open convex neighborhood of contained in (and hence ) and a -compact neighborhood of contained in (and hence ). ∎
Every locally compact and locally convex closed preordered space is weakly convex.
Let be a normally preordered space, a compact subset, a closed decreasing set on and a closed increasing set on . Further, let be an open and convex set on (not necessarily contained in ) such that
then there are an open decreasing set on and an open increasing set on , such that , , and
Since is a subspace and the -preorder property is hereditary, the subset , with the induced preorder and topology, is a -preordered space and, being compact, it is a normally preordered space (14).
By lemma 2.4 and by preorder normality of there are , open decreasing sets on and , open increasing sets on such that and
The set is closed on and hence compact on both and , decreasing on and disjoint from thus, where is closed decreasing on and is closed increasing on . By preorder normality of there are open decreasing on and open increasing on , such that
Analogously, is closed on , hence compact on both and , increasing on and disjoint from , is closed increasing in , is closed decreasing on , we have and we find open decreasing on , open increasing on such that
Let us define the open subsets of
We have because and if then while if we have . Analogously, .
Let us prove that . If then there are and , such that . The possibility is excluded because , and . Analogously, is excluded because and . Thus
Since which is convex we obtain , that is .
Using Prop. 2.2, since is open in and is decreasing in there is open decreasing on such that and applying preorder normality of there is , open decreasing on , such that . Analogously, we find open increasing sets on , such that . We have
Let be a -preordered -space, , and let be an open and convex neighborhood of . Then there are an open decreasing set and open increasing set , such that .
We already know that is normally preordered (14). Since is convex . Let , , be an admissible sequence for the -space . Without loss of generality we can assume . Each endowed with the induced topology and preorder is a compact -preordered space.
Let , . We have that is closed decreasing in , is closed increasing in and . Since is compact, by lemma 2.10 (with ) we can find , open decreasing set in , and , open increasing set in , such that , where and are the closed-hull maps of . Observe that and being closed subsets of are compact in . We define and , where is clearly closed decreasing in , and is closed increasing in . We have . We can proceed applying again lemma 2.10 with . Thus proceeding inductively, given , we find , respectively open decreasing and open increasing subsets of such that , , , and define and .
Note that and analogously, .
Let us define the sets and . The set is open because , and the set is open in so that, since for , , is open in and so is the union . The -space property implies that is open. Analogously, is open.
Let us prove that is increasing. Let then there is some such that . Let , then we can find some such that . Since , , and since is increasing on , thus . Analogously, is decreasing. Finally, if then there are some such that and setting , thus
As an immediate consequence we obtain the desired result.
Every weakly convex -preordered -space is a convex normally preordered space (and hence quasi-uniformizable).
Actually we proved something more, namely that a -preordered -space which is weakly convex at is convex at . Thus, by Prop. 2.8, in a -preordered -space , if local convexity and local compactness hold at , then convexity holds at .
Every locally convex -preordered locally compact -compact space is a convex normally preordered space (and hence quasi-uniformizable).
Every locally compact -compact space is a -space, and under local compactness local convexity and weak convexity are equivalent (Cor. 2.9). ∎
3 Convexity of -preserving spaces
The next definition is inspired by the property of global hyperbolicity in Lorentzian geometry, see Sect. 1.1.
A -preordered space is -preserving if every compact set has a compact convex hull .
Let be a -preordered space. If the topology does not distinguish the points of (e.g. if is locally convex at or antisymmetry holds at ) and admits a -compact neighborhood, then admits a base of -compact neighborhoods and, moreover, is weakly convex at .
Let be a -compact neighborhood of and hence , and let be an open neighborhood of and hence which we can assume contained in . We have to show that there is a compact -set neighborhood of such that , and analogously in the convex open neighborhood case. It suffices to apply lemma 2.6 with , . Observe that by local convexity (lemma 2.7) or antisymmetry at , if is any open neighborhood of we have . ∎
Every -preordered -preserving -space is convex at every point such that (i) the topology does not distinguish different points of [x], (ii) local compactness holds at (e.g. wherever it is locally compact and antisymmetric).
In particular, every -preserving -ordered locally compact -compact space is convex (and hence quasi-uniformizable).
Actually the -preserving property could be dropped provided we replace (ii) with the requirement that the point has a c-compact neighborhood, or that local compactness holds at and the -preserving property holds locally.
4 Compactly generated -preorders
In this section we study sufficient conditions for local convexity. The main idea is to consider preorders which, intuitively, are generated by relations which are limited, in the sense that do not connect arbitrarily ’far away’ points (compactness is used to give a rigorous meaning to this concept). Thus we shall be basically concerned with topological preordered spaces for which both topology and preorder are generated from local information.
For this type of preorder and for a locally compact space, given two related ‘far away’ points there is some point , , at ‘reasonable distance’ but not too close to the original point . From that it is possible to show that if local convexity is violated at then, by a limiting argument, some point exists such that and hence antisymmetry is violated at . This strategy has been used in mathematical relativity theory to prove that the relation (the smallest closed preorder containing the causal relation ) is locally convex ((13), Lemma 16) ((12), Lemma 5.5).
A -preordered space is a
--preordered space (read ’compactly generated
-preordered space’) if there is a relation
for every compact set the set is compact,
the preorder is the smallest closed preorder containing .
We shall also say that is a compactly generated preorder.
Note that in (ii) the smallest closed preorder exists because the family of closed preorders containing is non-empty as is a closed preorder which contains . Note that if satisfies (i)-(ii) then also satisfies them, thus can be chosen reflexive.
For applications in which is locally compact it is useful to observe that the condition
every point admits a closed and compact neighborhood such that is compact,
implies (i), and thus a space satisfying (i’) and (ii) is compactly generated. Note that if satisfies (i’)-(ii) then also satisfies them, thus can be chosen reflexive.
If is a -preordered compact space, then is compactly generated.
The conditions in the definition of compactly generated preorder are satisfied taking . ∎
The next result is worth mentioning although we shall not use it.
Let be a --preordered space, and let be a reflexive relation as in definition 4.1. The set of continuous isotone functions for coincides with the set of continuous isotone functions for .
If is a continuous isotone function for and we have, since , thus is a continuous isotone function for . If is a continuous isotone function for the relation is a closed preorder containing thus which implies , that is, is a continuous isotone function for . ∎
This result is interesting because in those cases in which is also normally preordered (the -space condition suffices (14)) this set of continuous isotone functions for allows us to recover , that is, iff for all continuous isotone functions , we have .
It is worth to mention a recent work by Akin and Auslander on recurrence problems and compactifications in dynamical systems (18). This section is very much related with their work, although we followed a different line of reasoning inspired by results in topological preordered spaces and relativity theory. In their paper they assume that is a separable locally compact metric space ((18), p. 50), while in our work second countability and Hausdorffness are not assumed, and local compactness is used only where it is strictly needed. We do not use compactification arguments as in their article.
We usually work with a reflexive relation because this is the interesting case from the topological point of view, as the elements of a quasi-uniformity contain the diagonal. Furthermore, the application to cone structures seems to require a reflexive . Observe that if is reflexive then the generalized recurrent set mentioned in ((18), Theor. 11) is the whole space. Our theorem 4.14 will be similar but stronger than their ((18), Theor. 14).
We find that our terminology concerning compactly generated -preordered spaces is more appropriate, since relations do generalize functions but the term proper is used for maps such that the inverse images of compact subsets are compact, while we do not take any inverse here. Maps which send compact set to compact sets are sometimes called compact. Finally, observe that our terminology places the accent on rather than . In applications there is often a natural choice for but, mathematically, it could be chosen with some freedom.
4.1 Some examples of compactly generated preorders
Most closed preorders appearing in applications are compactly generated. We give some examples proving conditions (i)-(ii) or (i’)-(ii) of remark 4.2.
Let us recall that in a spacetime (see Sect. 1.1) the relation is by definition the smallest closed and transitive relation containing . Let be a locally finite closed and compact covering of (it exist because of local compactness and ((30), Theor. 20.7)) and let . Since each is intersected only by a finite number of ,