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Higher Separation Axioms, Paracompactness And Scientiae Mathematicae Vol.2, No. 3(1999), 321{335 321 HIGHER SEPARATION AXIOMS, PARACOMPACTNESS AND DIMENSION FOR SEMIUNIFORM CONVERGENCE SPACES GERHARD PREU DedicatedtoProfessor Lamar Bentley on his 60th birthday Received July 23, 1999 Abstract. Higher separation axioms, paracompactness and dimension are de ned for semiuniform convergence spaces. Under the assumption that subspaces are formed in the construct SUConv of semiuniform convergence spaces one obtains the following 0 0 results: 1 Subspaces of normal (symmetric) top ological spaces are normal, 2 sub- 0 spaces of paracompact top ological spaces are paracompact, and 3 for some dimension functions, the dimension of a subspace is less than or equal to the dimension of the original space, where these dimension functions coincide with the (Leb esgue) covering dimension for paracompact top ological spaces. Additionally, Urysohn's Lemma, Tietze's extension theorem and some other extension theorems are studied not only in the realm of top ological spaces. Last but not least the interrelations b etween nearness spaces and semiuniform convergence spaces b ecome apparent. 0. Intro duction It is well{known that the construct Top of top ological spaces has several de ciencies, e.g. there do not exist natural function spaces in general (i.e. Top is not cartesian closed) and quotients are not hereditary (cf. [7; Thm. 2]). Furthermore, the formation of subspaces in Top is not satisfactory as the following example shows: Though a p oint or the closed pointinterval [0; 1] are distinguishable (they are non{homeomorphic), the top ological spaces obtained from the real line IR by removing a p ointor[0; 1] are not distinguishable (they are homeomorphic). Additionally, uniform concepts such as uniform continuity, completeness or uniform convergence cannot b e explained in the framework of top ological spaces. In ConvenientTop ology (cf. [16]) all these de ciencies are remidied byintro ducing semiuni- form convergence spaces (cf. [15]). In particular, one obtains non{isomorphic spaces in the ab ove example provided that subspaces are formed in the construct SUConv of semiuni- form convergence spaces instead of Top. This b etter b ehaviour of subspaces in SUConv leads to much nicer results even for classical top ological concepts such as paracompactness, normalityorcovering dimension as will b e explained in the following: According to Katetov [10] lter spaces can b e describ ed in the framework of merotopic spaces. On the other hand, lter spaces have also b een describ ed in the realm of semiuniform 1991 Mathematics Subject Classi cation. 54A05, 54A20, 54E05, 54E15, 54E17, 54B05, 54D15, 54D18, 54F45, 54F60, 18A40. Key words and phrases. Complete regularity, normality, full normality, paracompactness, dimension, semiuniform convergence spaces, lter spaces, convergence spaces, top ological spaces, merotopic spaces, nearness spaces, uniform spaces, maps into spheres, bire ective sub categories. 322 GERHARD PREU convergence spaces (cf. [15]). Thus, they form the link b etween semiuniform convergence spaces and merotopic spaces. From this fact pro t the de nitions of complete regularity, normality, full normality (resp. paracompactness) and dimension for semiuniform conver- gence spaces whichareintro duced in this article (complete regularity of lter spaces has b een studied earlier in [2] and [3]). It turns out that all higher separation axioms stud- ied here are hereditary and that for the small and the large lter{dimension intro duced here, the dimension of a subspace is less than or equal to the dimension of the original space, where these dimension functions coincide with the (Leb esgue) covering dimension for paracompact top ological spaces. Additionally, Urysohn's Lemma and Tietze's extension theorem are stated for normal lter spaces, and an extension theorem for Cauchy continuous (resp. uniformly continous) maps into spheres is related to the small lter dimension (resp. small uniform dimension) of normal lter spaces (resp. uniform spaces). Nearness spaces intro duced by Herrlich [6] form an imp ortant sp ecial case of merotopic spaces and several results on them are needed for the ab ovementioned theorems on semi- uniform convergence spaces. The terminilogy of this article corresp onds to [1] and [13]. 1. Preliminaries For the convenience of the reader some basic de nitions are rep eated. A semiuniform convergencespace is a pair (X; J ), where X is a set and J a set of lters X X 0 0 on X X such that 1 x_ x_ 2J for each x 2 X (wherex _ = fA X : x 2 Ag), 2 X 0 1 1 G2J whenever F2J and FG,and3 F2J implies F = fF : F 2Fg2J X X X X 1 (where F = f(y; x): (x; y ) 2 F g). A map f :(X; J ) ! (Y; J )between semiuniform X Y convergence spaces is called uniformly continuous provided that (f f )(F ) 2J for each Y F2J . The construct of semiuniform convergence spaces is denoted by SUConv. X A uniform space (X; V ) in the sense of Weil [21] may b e considered to b e a semiuniform convergence space (X; J )withJ =[V ]where[V ]=fF 2 F (X X ): FVgand X X F (X X ) denotes the set of all lters on X X . In this case (X; J )isalsocalleda X principal uniform limit space. A symmetric top ological space (= R {space) (X; X )maybe 0 considered to b e a semiuniform convergence space (X; J ), where J = fF 2 F (X X ): X X there is some x 2 X with FU (x) U (x)g and U (x) denotes the neighb orho o d lter X X X of x 2 X w.r.t. X . In this case (X; J ) is called a topological semiuniform convergence X space. Every semiuniform convergence space (X; J ) has an underlying lter space (X; ), X J X where = fF 2 F (X ): FF 2 J g and F (X ) denotes the set of all lters on X J X X (rememb er that a lter space is a pair (X; ), where X is a set and F (X ) contains all lters x_ with x 2 X and for each F 2 , all lters G on X with F G;amap 0 0 f :(X; ) ! (X ; )between lter spaces is called Cauchy continuous provided that 0 f (F ) 2 for each F2 ). Furthermore, a lterspace (X; )may b e considered to b e a semiuniform convergence space (X; J )whereJ = fF 2 F (X X ) : there is some G2 with GG Fg. In particular, a semiuniform convergence (X; J ) is called Fil{determined X provided that J = J . X J X A symmetric Kent convergencespace isapair(X;q )whereX is a set and q F (X ) X 0 0 suchthat1 (_x;x) 2 q for each x 2 X ,2 (G ;x) 2 q whenever (F ;x) 2 q and GF, T 0 0 3 (F\x;_ x) 2 q whenever (F ;x) 2 q ,and4 (F ;x) 2 q and y 2 fF : F 2Fg imply HIGHER SEPARATION AXIOMS 323 0 0 (F ;y) 2 q (symmetry condition). A map f :(X; q ) ! (X ;q )between (symmetric) Kent 0 convergence spaces is called continuous provided that (f (F );f(x)) 2 q for each(F ;x) 2 q . The construct of symmetric Kent convergence spaces is denoted by KConv .Every semi- S uniform convergence space(X; J ) has an underlying symmetric Kent convergencespace X .Furthermore, a symmetric Kentcon- i F\x_ 2 ) de ned by(F ;x) 2 q (X; q J X J J X X ) vergence space (X; q )may b e considered to b e a semiuniform convergence space (X; J q where = fF 2 F (X ): there is some x 2 X with (F ;x) 2 q g. q For every symmetric Kentconvergence space (X; q ) a closure op erator cl is de ned by q cl A = fx 2 X : there is some F2F (X ) with (F ;x) 2 q and A 2Fgfor each subset A of X ; q o ccasionally, one writes A instead of cl A. In particular, X = fO X : cl (X nO )= X nO g q q q is a top ology on X .For each top ological space (X; X ), de ne q F (X ) X by(F ;x) 2 q X X i FU (x). A symmetric Kent convergence space (X; q ) is called topological provided X that q = q . X q A subset A of a semiuniform convergence space (X; J ) is called closed (resp. dense) X provided that A = cl A (resp. cl A = X ). A semiuniform convergence space (X; J ) q q X J J X X ) is called regular provided that for each F2J the sub lter F generated by the lter X F : F 2Fgb elongs to J , where F denotes the closure of F in the underlying base f X Kent convergence space of the pro duct space (X; J ) (X; J ) (note that SUConv is X X a top ological construct!). The underlying symmetric Kentconvergence space (X; q )of J X a regular semiuniform convergence space (X; J ) is regular, i.e. for each(F ;x) 2 q , X J X ( F ;x) 2 q where F is generated by fF : F 2Fgand F = cl F ; furthermore, q J X J X a symmetric top ological space (X; X ) is regular in the usual sense i it is regular as a semiuniform convergence space. Using [5; 3.7.13. and 4.12.6.] one obtains the following Theorem. Let (X; J ) be a principal uniform limit space (resp. a topological semiuniform X convergencespace). Then each uniformly continuous map f :(A; J ) ! (Y; J ) from A Y a dense subspace (A; J ) of (X; J ) to a complete, regular and separated semiuniform A X f :(X; J ) ! convergencespace (Y; J ) has a unique uniformly continuous extension X Y (Y; J ).
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