aa r X i v : . [ m a t h . GN ] D ec A remark on nearness spaces
Jan-David Hardtke
Abstract.
We give a proof of the well-known fact that the cate-gory of nearness spaces is bireflective in the category of merotopicspaces which uses Zorn’s Lemma instead of the usual constructionby transfinite induction.
First let us introduce some notation and definitions. If X is any set wedenote by P ( X ) the power-set of X and by c ( X ) the set of all non-emptycoverings of X . For A , B ∈ c ( X ) we say that A is a refinement of B (or A refines B ), denoted by A ≺ B , if for every A ∈ A there is B ∈ B such that A ⊆ B . We further put A ∧ B = { A ∩ B : A ∈ A , B ∈ B} , which is obviouslyagain a covering of X . For any unexplained notions from category theorywhich we will use in the sequel the reader is referred to [6] and [7].Now recall that a merotopic space is a pair ( X, µ ), where µ ⊆ c ( X ) isnon-empty and such that the following holds:(i) A ∈ µ, B ∈ c ( X ) and A ≺ B ⇒ B ∈ µ (ii) A , B ∈ µ ⇒ A ∧ B ∈ µ Then µ is sometimes called a merotopic structure on X and the elements of µ are called uniform coverings.A map f : X → Y between merotopic spaces ( X, µ ) and (
Y, ν ) is calleduniformly continuous (with respect to µ and ν ) if f − [ A ] ∈ µ for every A ∈ ν , where f − [ A ] := (cid:8) f − ( A ) : A ∈ A (cid:9) .Clearly, the merotopic spaces together with the uniformly continuousmaps as morphisms form a concrete category over the category of sets, i. e.a construct, which will be denoted by Mer .For a merotopic space (
X, µ ) and a subset A ⊆ X we define the interiorof A with respect to µ as int µ ( A ) = { x ∈ X : { A, X \ { x }} ∈ µ } .It is well-known and easily checked that the following assertions hold(cf. [6, Proposition 3.2.2.3], where merotopic spaces are called semi-nearnessspaces):(a) int µ ( A ) ⊆ A ∀ A ⊆ X Keywords : merotopic spaces; nearness spaces; bireflective subcategory; Zorn’s Lemma
AMS Subject Classification (2010): 54E17; 54B30 µ ( X ) = X, int µ ( ∅ ) = ∅ (c) A ⊆ B ⊆ X ⇒ int µ ( A ) ⊆ int µ ( B )(d) int µ ( A ∩ B ) = int µ ( A ) ∩ int µ ( B ) ∀ A, B ⊆ X A merotopic space (
X, µ ) is called a nearness space if { int µ ( A ) : A ∈ A} belongs to µ for every A ∈ µ . Then µ is also called a nearness structureon X . The nearness spaces (and uniformly continuous maps) induce a fullsubconstruct of Mer which will be denoted by
Near .The merotopic spaces were originally introduced by Katˇetov in [5], thoughnot in the formulation above but in an equivalent version using so called mi-cromeric collections of subsets of X instead of uniform coverings. One canalso use a concept of “near” collections of subsets of X and an associatedclosure operator for equivalent definitions of merotopic and nearness spaces.In such a way the nearness spaces were originally introduced by Herrlich in[2] and [4]. For details on the various equivalent formulations we refer thereader to [4].It is also due to Herrlich that Near is bireflective in
Mer . This theoremis usually proved by constructing the bireflective modification of a givenmerotopic space with respect to
Near via transfinite induction (cf. [4,Theorem 8.1] or [6, Theorem 3.2.2.5]).We want to give a different proof here, which is based on Zorn’s Lemma.We begin with an easy lemma.
Lemma. If µ and µ are two nearness-structures on the set X then µ = {A ∈ c ( X ) : ∃A ∈ µ , ∃A ∈ µ such that A ∧ A ≺ A} is again a nearness-structure on X that contains µ and µ . Moreover, everymerotopic structure on X that contains µ and µ must also contain µ . Note that if we already knew that
Near is bireflective in
Mer this lemmawould be an immediate consequence of the general way of constructing initialobjects in
Mer (cf. [6, Remark 3.2.2.2 2 and Theorem 3.2.2.1]), but sincewe want to use it to show the bireflectivity result we have to give a directproof, which can be easily done as follows.
Proof.
Obviously we have µ , µ ⊆ µ and ( X, µ ) is easily seen to be a mero-topic space. Now pick any
A ∈ µ . By definition there are A ∈ µ and A ∈ µ such that A ∧ A ≺ A . It follows that A ′ i = { int µ i ( A ) : A ∈ A i } ∈ µ i for i = 1 , A i ∈ A i arbitrarily for i = 1 ,
2. If x ∈ int µ ( A ) ∩ int µ ( A ) then { A i , X \ { x }} ∈ µ i for i = 1 , { A , X \ { x }} ∧ { A , X \ { x }} ≺{ A ∩ A , X \ { x }} it follows that x ∈ int µ ( A ∩ A ).Thus we have int µ ( A ) ∩ int µ ( A ) ⊆ int µ ( A ∩ A ) and because of A ∧A ≺A we find A ∈ A such that A ∩ A ⊆ A and hence int µ ( A ) ∩ int µ ( A ) ⊆ µ ( A ). So we have that A ′ ∧ A ′ is a refinement of { int µ ( A ) : A ∈ A} andhence the latter set belongs to µ which shows that ( X, µ ) is indeed a nearnessspace. The “moreover” part is clear.Now we are ready to prove the bireflectivity of
Near in Mer . Theorem. Near is bireflective in
Mer .Proof.
Let (
X, µ ) be any merotopic space and put M = { ν ⊆ c ( X ) : ( X, ν ) ∈ Near and ν ⊆ µ } . The set M is partially ordered by inclusion and M is non-empty, because {A ⊆ P ( X ) : X ∈ A} is an element of M .If S is any non-empty chain in M then it is easy to see that S S is again in M and hence by Zorn’s Lemma there is a maximal element ˜ µ of M .Since ˜ µ ⊆ µ the identity map id X : ( X, µ ) → ( X, ˜ µ ) is uniformly continuous.Now if ( Y, ν ) is another merotopic space and f : ( X, µ ) → ( Y, ν ) is uniformlycontinuous we can put µ f = (cid:8) B ∈ c ( X ) : ∃A ∈ ν with f − [ A ] ≺ B (cid:9) and show exactly as in the proof from [6, Theorem 3.2.2.5] that ( X, µ f ) is anearness space. Next we define¯ µ = {A ∈ c ( X ) : ∃A ∈ ˜ µ, ∃A ∈ µ f such that A ∧ A ≺ A} . By the preceding lemma ( X, ¯ µ ) is a nearness space and ˜ µ, µ f ⊆ ¯ µ . Since f is uniformly continuous with respect to µ and ν it follows that µ f ⊆ µ and because ˜ µ is also contained in µ it follows that ¯ µ ⊆ µ , in other words¯ µ ∈ M and by the maximality of ˜ µ we must have ¯ µ = ˜ µ . Hence µ f ⊆ ˜ µ which implies that f : ( X, ˜ µ ) → ( Y, ν ) is uniformly continuous. Thus wehave shown that ( X, ˜ µ ) is our desired bireflective modification of ( X, µ )with respect to
Near . References [1] J.W. Carlson,
Topological properties in nearness spaces , Gen. Top. Appl. (1978),111–118.[2] H. Herrlich, A concept of nearness , Gen. Top. Appl. (1974), 191–212.[3] , On the extendibility of continuous functions , Gen. Top. Appl. (1974), 111–118.[4] , Topological structures , Math. Centrum Amsterdam (1974), 59–122.[5] M. Katˇetov, On continuity structures and spaces of mappings , Comment. Math. Univ.Carolinae (1965), no. 2, 257–278. [6] G. Preuß, Theory of Topological Structures : An Approach to Categorical Topology ,Mathematics and its Applications, D. Reidel Publishing Company, Dordrecht–Boston–Lancaster–Tokyo, 1988.[7] ,
Foundations of Topology : An Approach to Convenient Topology , Kluwer Aca-demic Publishers, Dordrecht, The Netherlands, 2002.
Department of MathematicsUniversit¨at LeipzigAugustusplatz 10, 04109 LeipzigGermany
E-mail address: [email protected]@math.uni-leipzig.de