Pre-Hausdorff Spaces

1 Pre-Hausdorff Spaces JA Y STINE Dep artment of Mathematics, Miseric or dia University, Dal las, P A, 18612 , U.S.A. M. V. MIELKE Dep artment of Mathematic s, U niversi t y of Miami , Cor al Gables, FL, 33124, U.S.A. Abstract. This pap er introduces three sepa ration conditions for top olog- ical spaces, called T 0 , 1 , T 0 , 2 (”pre-Hausdor ff”), and T 1 , 2 . These conditions generalize the classical T 1 and T 2 separation axioms, and they have adv an- tages over them top ologically which we discuss. W e establish several different characterizations of pre-Hausdorff spaces, and a characterization o f Hausdor ff spaces in terms of pre-Haus do rff. W e also discuss so me classi cal Theorems of general topolog y which can or ca nnot b e generalized by replacing the Hausdorff condition by pre-Ha us do rff. Mathematics Sub ject Classifications (2000): 18B30 , 54A05, 54D10. Keyw ords: topo logical separation prop erties, top o logical category , left adjoint, sober spa ce In tro duction The notion of separa tion is fundamen ta l to top ology . Ev en so, the c la ssical separation axio ms (T 0 , ..., T 4 ) are so metimes o verlo ok ed in a first co urse or, alternatively , some consider the T 2 axiom (for instance) as b eing sufficien tly weak that all spaces are a ssumed T 2 and no further consideration is g iv en to separation. While this may b e r easonable in some settings, it is certainly not in others . Analysis often takes place in the setting o f metric spaces, which are T 2 , whereas geometry often uses pseudometric (more gener a lly , unifor m) spaces which are not nec e ssarily T 2 . Herrlich argues (in [7]) that ” there are s ufficien t reasons for top ologists to pa y serious atten tion to non-Hausdorff spaces ... finite Hausdorff spaces ar e rare and not v e ry in teres ting ... a 14- elemen t set carr ies just a single Hausdorff top ology but 98,4 84,324,25 7 ,128,207,032,183 T 0 topo logies”. In this pap er w e define a genera lized Hausdorff separ a tion c o ndition ca lled pre- Hausdorff, w hich is sa tisfied b y many impo rtan t non-Hausdor ff s paces. In [15 ] it is shown that a uniform space is T 0 if and only if it is T 2 , and the pro of of this r ev ea ls that all uniform spaces are pr e-Hausdorff. F ollowing Her r lic h’s argument ab o ve, the worthiness of studying pre-Hausdorff s paces can b e justified b y their abundance: a 14- e le ment set c a rries 19 0,899,322 distinct pre - Hausdorff topo logies (see C o rollary 2.3 ). The pap er is organized a s follows. Section 1 contains definitions of three separation axio ms for top ological space s and exa mples to show ho w they are 1 related. W e pr o ve that the categor ies for med b y the spaces whic h satisfy these axioms ar e top ological categories a nd, further , that these categories are reflective in the ca tegory o f topolog ical spaces. In section 2, we g iv e several characterizations of pre-Haus do rff spaces in terms of Hausdorff separation and some e q uiv alence relations. Finally , in section 3, we consider some classical Theorems of g eneral top ology which can or cannot b e genera lized b y replacing the Ha usdorff condition by pre-Hausdorff. Throughout the paper, TOP will be used to denote the ca tegory of topo - logical spaces and conti nu ous functions. F or i = 0,1,2, T i -TOP will denote the full sub categories of T O P consisting of the T i spaces (see [4 ] , page 138).. 1. T i,j - Spaces DEFINITION 1.1. A top ological space X is c a lled a T i,j - spa ce (for 0 ≤ i < j ≤ 2) if and only if each pair of p oin ts a, b ∈ X which has a T i - separa tion in X also has a T j - sepa ration in X. NOT A TION: The categor ies consisting of the T i,j - spaces, a long with con- tin uo us functions, will be denoted T i,j - TOP . EXAMPLE 1.2. T 0 , 2 spaces hav e been r efered to as pre-Hausdorff spaces in the liter ature (see[14]). EXAMPLE 1.3. T 0 , 1 spaces ha ve b een refered to as R 0 - spaces in the literature (see[8]). An R 0 - space is a topolog ical space X which satisfies: x ∈ { y } (the topolo gical clo sure of { y } ) if and only if y ∈ { x } , for all pairs of points x, y ∈ X. Evidently then, ev ery neighborho o d o f x con ta ins y if and only if every neighborho o d of y contains x. Now if x a nd y hav e no T 0 separation, then this condition is satisfied. On the o ther hand, if x and y do hav e a T 0 separation, say x has a neighbo rhoo d not con taining y , then y m ust have a neighbo rhoo d not containing x; i.e., x and y must ha ve a T 1 separation. Thus R 0 - spaces are exactly the T 0 , 1 - spaces. EXAMPLE 1.4. Clearly T j spaces ar e alwa ys T i,j , but T i spaces need not be T i,j . A Sier pinski s pace (i.e., a t wo-p o in t set , say X = { 0,1 } , with one prop er op en s et, say { 1 } ), for instance, is T 0 , but neither T 0 , 1 nor T 0 , 2 ; while a T 1 space whic h is no t T 2 will no t be T 1 , 2 . F urthermore, a T i,,j space need not be either T i or T j . An indiscrete space with more than o ne element, for instance, is T i,,j for ea c h i, j , but is not T 0 . EXAMPLE 1.5. Clearly T 0 , 2 spaces ar e both T 0 , 1 and T 1 , 2 . How ever, a T 1 , 2 space need not b e T 0 , 1 (and, consequently , not T 0 , 2 either), as in the case of a Sierpinski space, for example. F urthermor e, a T 0 , 1 space ma y be neither T 0 , 2 nor T 1 , 2 . This is the c ase if, for example, a s pace is T 1 but no t T 2 . The follo wing Theorem shows that the categor ies T i,j - TOP hav e a desirable prop ert y that is not shared by the ca tegories T i - TO P . 2 THEOREM 1.6 . The ful l sub c ate gories T i,j - TOP ar e t hemsel ves t op olo gic al over SET (the c ate gory of sets and fun ctions). Mor e over, their inclusions into TOP pr eserve initial lifts and, c onse quently they pr eserve al l limi t s. Pr o of: W e prove the Theo r em for T 0 , 2 -TOP , the ca ses T 0 , 1 - TOP and T 1 , 2 - TOP being similar. Clea rly the restriction of the forgetful functor U : TOP → SET is b oth concrete and has s et-theoretic fibers. So w e show tha t the structure induced o n a set from an a rbitrary family of T 0 , 2 spaces yields a T 0 , 2 space. This will a lso show that initial lifts in T 0 , 2 - TO P a re computed as they are in TOP and, thus, the inclusion functor preser v es them. Supp ose that (X, τ ) is the induced top ological space on a set X from a family { (X i , τ i ) } i ∈ I of T 0 , 2 spaces via a family of functions { f i : X → X i } i ∈ I . F urther supp ose that x, y ∈ X hav e a T 0 - separation in τ by , say , U x ∈ τ , where x ∈ U x and y / ∈ U x . W e can as sume that U x is a basis element o f τ s o that U x = n T j =1 f -1 i j (V j ), where each V j is open in X i j for each j = 1, 2 , ..., n . Then ∃ k , 1 ≤ k ≤ n , with f i k (x) ∈ V k and f i k (y) / ∈ V k ; i.e., f i k (x) and f i k (y) ha ve a T 0 - separation in X i k . Since X i k is T 0 , 2 , ∃ nbhd s. U a nd W of f i k (x) and f i k (y) (r esp.) such that U ∩ W = ∅ . There fo re (X, τ ) is T 0 , 2 . COROLLAR Y 1.7. The inclusion functors inc i,j : T i,j - TOP → TOP ea c h hav e a left a djoin t L i,j . Pr o of: Note that any indiscrete space with tw o element s forms a small (one element ) co generating set for any o f the categor ies T i,j - TOP . Since the functor s inc i,j are contin uous by Theo rem 1.6, the r esult follows immediately from the Corollar y o n page 126 of [12]. Note: An explicit description of the left adjoint to inc 0 , 2 is given b elow, in the discussion following Theorem 2.18 . Another description of L 0 , 2 using transfinite recursio n can b e found in [19], where this a pproac h is then adapted to give a n explicit descr iption o f the functor L 0 , 1 : TO P → T 0 , 1 - TOP . It is also shown there that these left adjoints are retra ctions. 2. Pre-Hausdorff Spaces This section is c o ncerned specifica lly with T 0 , 2 -TOP , the catego r y o f pre- Hausdorff spaces . In [18] Steiner defines principal topolo gies in terms of ul- tratop ologies, and prov es that a top o logical space is pr incipal if and o nly if arbitrary in ter s ections of op en sets are op en (such spaces are also refered to as Alexandroff spaces in the literature, see [2] ; how ever the term Alexandroff space also app ears in a different context, see [3] ). The following result will b e used to ga in insig h t into finite pre- Hausdorff spa ces, and to coun t the n umber o f distinct pr e - Hausdorff top ologies on a a finite set. W e note that in this result, as in [11] pag e 11 3, w e make a distinction b et ween regular spaces and T 3 spaces; namely , that r egular spaces need no t hav e closed po in ts. THEOREM 2.1. Su pp ose (X, τ ) is a princip al sp ac e (X, τ ). The fol lowing ar e e qu iva lent : 3 (i) (X , τ ) is pr e-Hausdorff. (ii) (X , τ ) is r e gular. (iii) (X, τ ) has dimension 0; i.e., has a b asis c onsisting of clop en sets (se e [9] , p age 10, B). (iv) The t op os of she aves on X is Bo ole an; i.e., the ne gation op er ator ¬ : τ → τ satisfies ¬¬ = id (se e [13] , p age 270). Pr o of: (i) ⇐ ⇒ (ii) Supp ose A ⊂ X is clo sed and p ∈ A C . Then p has a T 0 - sepa ration from each p oin t a ∈ A. If X is pre-Hausdor ff, then ( ∀ a ∈ A)( ∃ N a , N p a ∈ τ ) such that a ∈ N a , p ∈ N p a , and N a ∩ N p a = ∅ . Then p ∈ U = \ a ∈ A N p a , A ⊂ V = [ a ∈ A N a , and U ∩ V = ∅ . Since X is principa l we ha ve that U is o pen and, consequently , X is reg ular. Co n versely , supp ose that x, y ∈ X hav e a T 0 - separa tion b y , say , U x ∈ τ , where x ∈ U x and y / ∈ U x . Then U C x is closed so, if X is regula r, there are disjoint o p en sets U and V such that x ∈ U and U C x ⊂ V. Thu s, X is pr e-Hausdorff. (ii) ⇐ ⇒ (iii) See [2] , Theorem 2.9. (iii) ⇐ ⇒ (iv) In the top os of sheav es on X, the negation op erator ¬ : τ → τ is defined b y ¬ U = interior (U C ) (see [13], Chapter 2). Then ¬¬ U = interior( U), and so ¬¬ U = U iff U = in terio r(U); i.e., iff U is a regular o pen set (see [4 ], page 92 ). It is e asily shown that a n op en set is regula r iff it is clo p en. REMARKS 2.2. (i) The pro of o f Theorem 2.1 s ho ws tha t a regular space is pr e-Hausdorff even if it is not principal. Clearly the conv ers e is false; for if X is a Hausdorff space which is no t T 3 , then X is pre-Hausdor ff but not regula r. (ii) Also, a 0-dimensio nal space is pre-Hausdo rff even if it is not principal. How ever this is no t true con versely; for the set o f r eal n umbers R with the usual op en interv al top ology is a (pre-) Ha usdorff space which is not 0-dimensional. In fact, dim( R) = 1 (see [ 9 ] , page 25, E xample I II). In [5] it is shown that a principal top ological spa ce (X, τ ) is regula r if and only if the minimal basis for τ forms a partition of X. Consequently , we have the immediate COROLLAR Y 2 .3 . If X is a finite set, t hen t he distinct pr e-Hausdorff top olo- gies on X ar e in one-to-one c orr esp ondenc e with the distinct p artitions on X. In [5] there is an alg orithm using matrices, a nd a computer program, to compute the num b er of r e gular (hence, pre-Hausdorff ) top ologies on a finite set. Alternatively , several methods for counting the num b er of pa rtitions on a set with n-elements, the so -called ” n-th Bell Num b er” B(n), a re w ell- kno wn (see [17] , , pa ge 33). The 1 4th Bell n umber, for instance, is B(14) = 190 ,899,322, which is accordingly the num ber o f distinct pre-Hausdorff topolog ies on a 14- element set as mentioned in the introduction. Finite (and other) pre-Hausdor ff spaces can a lso be describ ed using the no - tion of a Bo rel field. 4 DEFINITION 2.4. A Borel field F (on a fixed set B) is a no n-empt y family of subsets of B suc h that F is closed with resp ect to co mplemen ts a nd countable unions; i.e., F sa tisfies: (i) if A ∈ F, then A C ∈ F (ii) if { A i } ∞ i =1 ⊂ F, then ∞ [ i =1 A i ∈ F. REMARKS 2.5. (i) A Borel field is also known as a σ -algebra (see [16], page 17, for instance). (ii) If F is a Bo rel field o n B, then clearly B ∈ F a nd ∅ ∈ F.. (iii) It follows immediately fro m DeMorgan’s la ws that a Bor el field is als o closed with resp ect to co un table in tersections; i.e., if { A i } ∞ i =1 ⊂ F, then ∞ \ i =1 A i ∈ F. (iv) If F is a Bor el field on B a nd F is coun table, then (B, F) is a t op ological space which has the following pro perties: (1) arbitrar y intersections of op en sets are op en, and (2) every op en set is also clo sed; i.e., every op en set is clop en (bo th op en and closed). COROLLAR Y 2.6 . Supp ose X is a finite set and τ is a family of subsets of X. τ is a Bor el field if and only if (X, τ ) is a pr e-Hau sdorff sp ac e. Pr o of: F ollows immediately from Remark 2 .5 (iv) and Theorem 2 .1. Clear ly this result is also true for any set X if τ is co un table. In [2 0], Szekeres and Binet prov e that the set o f a ll B o rel fields on a finite set is in one-to-one corresp ondence with the num b er of equiv a le nce rela tions on that set. It is well known that the num be r of equiv alence r elations o n a finite set are in o ne-to-one corresp ondence with the num b er of partitions on that set. Consequently , Corollar y 2 .6 is equiv alent to Co rollary 2.3. Of the 190 ,8 99,322 distinct pre-Hausdorff top ologies on a set with 14 ele- men ts there are, of course, many whic h are homeomor phic. T o characterize homeomorphic pair s of finite pre-Hausdor ff s paces, we loo k at the basis con- sisting of the ” minimal” op en sets which, by the pro o f of Co rollary 2.3, forms a pa r tition. F o r a finite pre-Ha us dorff space X, we shall denote this partition which genera tes X by P X . The following sho ws that for finite pre-Hausdorff spaces to b e homeomor phic, their generating partitions must ”loo k” the s ame. PROPOSITION 2.7. Finite pr e-Hausdorff sp ac es (X , τ ) and (Y, σ ) ar e home omorphic if and only if ther e exists a bije ctive c orr esp ondenc e b etwe en P X and P Y that pr eserves the c ar dinality of the c orr esp onding blo cks. Pr o of: If X a nd Y a re homeomor phic, then the co ndition on their generat- ing partitions follows immediately since a homeomorphism is a bijectiv e op en mapping. Conv ersely , s uppose there exists a bijective corres pondence b et ween P X and P Y which preserves the cardinality of the corr esponding blo cks, and 5 that P X = { B i } k i =1 and P Y = { C i } k i =1 are labeled so that B i and C i each hav e the s ame car dinalit y for a ll i = 1, 2, ..., k . Then, for each i , we can choose a bijection f i : B i → C i . The function f: X → Y defined by f(x) = f i (x), for x ∈ B i , is clea rly a homeomorphism. It fo llows from Prop osition 2.7 that the num b er of non-homeomor phic pre- Hausdorff spaces on a set with n-elemen ts is p(n) = the n umber of partitions of n according to the following. DEFINITION 2.8. A pa rtition of a p ositiv e integer n is a finite nonincre a sing sequence o f po sitiv e integers λ 1 , λ 2 , ..., λ r such that r X i =1 λ i = n. The problem of computing p(n) in general is complex and has recieved muc h atten tion fr o m mathematicia ns , esp ecially after the landmar k pap er by G. H. Hardy a nd S. Rama n ujan in 19 18 ([6]). A comprehensive summary of res ults can b e found in [1], wher e there is also a table of v a lues for p(n) up to p(100 ). Thu s, the num b er of non-homeomor phic pre-Hausdo r ff top ologies on a set with 14 elements is p(14) = 13 5 (see [1 ] , page 238). Suppos e X is a topo logical space, and B ⊂ X. Recall that a p oin t b ∈ X is called a gener ic p oint o f B provided { b } = B, and that X is called sob er provided every closed irreducible (i.e., ca nnot be decomp osed into a unio n of tw o or mo r e smaller close d subsets) subset of X has a unique generic p oint . See [10] , page 230 for a n in ter esting explanation of the ter m sob er. THEOREM 2.9. A t op olo gic al sp ac e (X, τ ) is Hausdorff if and only if X is b oth pr e-H ausdo rff and sob er. Pr o of: The implication to the right is immediate beca use Hausdor ff spa c es are naturally pr e-Hausdorff and, furthermore, they are a lw ays sob er (see [13 ] , page 47 5). Now suppos e that X is b o th pre-Hausdo rff and sob er, but not Hausdorff. Then ∃ x, y ∈ X such that x 6 = y , and x and y ha ve no T 2 separation in τ . Then x and y have no T 0 separation in τ either, which implies { x } = { y } . But then { x } is a clo sed irreducible subset o f X with mo r e than o ne generic po in t. It is w ell- kno wn that a topolog ical space X is Hausdorff if and o nly if the diagonal ∆ X (= { (x,x) : x ∈ X } ) is clo sed in the pro duct space X 2 . Analog ous results for a pre- Hausdorff space (X, τ ) a re given in terms of the following r elation R 0 on X. R 0 = { (x, y) : x and y hav e no T 0 separation in τ } ⊂ X 2 Clearly R 0 is an equiv a lence relation. THEOREM 2.10. The fol lowing ar e e qu iva lent . 6 (i) X is pr e-Hausdorff. (ii) R 0 is close d in X 2 . (iii) R 0 = ∆ X . (iv) The quotient sp ac e X R 0 is H aus dorff . Pr o of: W e show that each of (ii), (iii), and (iv) is equiv alent to (i). (ii) If R 0 is closed and x, y ∈ X ha ve a T 0 separation in τ , then (x, y) / ∈ R 0 = R 0 . So ∃ U, V ∈ τ s uc h that (x, y) ∈ U × V and (U × V) ∩ R 0 = ∅ . But this implies that U and V ar e disjoint, for if p ∈ U ∩ V, then (p, p) ∈ (U × V) ∩ R 0 . Therefore x and y ha ve a T 2 separation, and X is pre-Hausdor ff. Conv er sely , if X is pre-Hausdor ff a nd (x, y) ∈ R C 0 , then x and y hav e a T 2 separation in τ by , say , N x and N y . Then (x, y) ∈ N x × N y ⊂ R C 0 , so R 0 is closed. (iii) Tha t (iii) implies (i) follows immediately from (ii). F or the r ev erse implication, w e hav e ∆ X ⊂ R 0 = R 0 (since X is pre-Hausdor ff ) so that ∆ X ⊂ R 0 . F or the reverse inclusion, supp ose a po in t (x, y) ∈ R 0 = R 0 has nbhd. U × V in X 2 . Then (x, x) ∈ U × V ∩ ∆ X , so that (x, y) ∈ ∆ X . (iv) Supp ose X R 0 is Hausdorff, and that distinct p oin ts x, y ∈ X hav e a T 0 separation by a nbhd. o f x. Then [x] (= { z: z R 0 x } ) 6 = [y], so that [x] and [y] hav e a T 2 separation in X R 0 . Since the canno nical map q: X → X R 0 is con tinuous, x and y hav e a T 2 separation in X. Conv ersely if X is pre-Hausdorff, then R 0 is closed b y (ii). F urthermore, q: X → X R 0 is easily seen to be an op en map. Consequently X R 0 is Hausdo rff (see [4] , 1 .6, page 1 4 0). W e now show that a n y space can b e universally r etracted o n to a Ha us dorff space in the sense o f adjunction as follows. LEMMA 2.11. If (X, τ ) is any top ological spa ce, (Y, σ ) is a T 0 space, and f: X → Y is contin uous, then f factors uniquely through the quo tien t map q: X → X R 0 ; i.e., ∃ ! contin uous f : X R 0 → Y such that f = f ◦ q. Pr o of: Define f ([x]) = f(x). Then f is well-defined since Y is T 0 , and f is contin uous since X R 0 is equipp ed with the coinduced top ology; i.e., the q uotien t topo logy in TOP . THEOREM 2.12. The inclusion functor inc 2 , 2 : T 2 -T op ֒ → T 0 , 2 -T op has a left adjoint L 2 , 2 which is a r etr act. Pr o of: Define L 2 , 2 (X) = X R 0 . By Lemma 2.11, the quo tien t map q: X → X R 0 provides a universal ar ro w from an y pre-Hausdor ff space X to the Hausdorff space X R 0 . The o b ject X R 0 and the universal arrow q completely determine the left adjoin t to inc 2 , 2 (see [12], Theorem 2 (ii), page 81). L 2 , 2 is a retract by Theorem 2.10 (iii). COROLLAR Y 2.13. The inclusion functor inc 2 : T 2 - T op ֒ → T op has a left adjoint L 2 which is a r etra ct . Pr o of: Com bining Cor ollary 1.7 with Theorem 2.1 2, we define L 2 = L 2 , 2 ◦ L 0 , 2 . 7 The functor L 2 can b e describ ed without the use of L 0 , 2 and L 2 , 2 . T o this end, we now construct L 2 directly b y wa y of forming quotien ts by an equiv alence relation. W e tak e a general appro a c h which also shows T 0 -TOP and T 1 - TOP to b e reflective, and gives an explicit description of the left adjoints to their inclusions into TOP . DEFINITION 2.14. Let (X, τ ) be a top ological space. F or each i = 0, 1, 2, define a r e la tion R i on X by: (x, y) ∈ R i iff ∀ Y ∈ T i - TOP , ∀ co n tinuous f : X → Y, f(x) = f(y). REMARK 2.15. R 0 as defined in 2.14 equa ls R 0 as defined a bov e. LEMMA 2.16. F or e ach i = 0, 1, 2 we have the fol lowing: (i) R i is an e quivalenc e r elation. (ii) If Y ∈ T i - TOP and f : X → Y is c ont inuous, then f factors thr ough the qu otient map q : X → X R i . (iii) X R i ∈ T i - TOP. Pr o of: (i) Straightforward. (ii) Giv en a contin uous function f : X → Y with Y ∈ T i - TOP , define f : X R i → Y by f([x]) = f(x). Then f is well-defined by definition of R i , and f is contin uous since X R i has the q uo tien t top ology in TOP . (iii) Supp ose that [x] 6 = [y] in X R i . Then ∃ Y ∈ T i - TOP and ∃ contin uous f : X → Y with f(x) 6 = f(y), whic h implies that f(x) and f(y) hav e a T i - separa tio n in Y. Then [x] and [y] hav e a T i - separa tion in X R i via inv erse image of f : X R i → Y. THEOREM 2.17 . F or e ach i = 0,1, 2, the inclusion functor inc i : T i - TOP ֒ → TOP has a left adjoi n t L i : TOP → T i - TOP. Mor e over, e ach L i is a r etr act. Pr o of: Define L i ((X, τ )) = X R i . Then, b y Lemma 2.16, L i ((X, τ )) ∈ T i -TOP , and the quotient map q : X → X R i is universal among all arr o ws from X in to a T i - spa ce. If X ∈ T i - TO P then, clea rly , L i (X) = X. The functor L 0 , 2 of Corollar y 1.7 can a lso b e explicitly describ ed using the equiv alence relation R 2 . Indeed, if (X, τ ) is a topo logical space, then X R 2 is Hausdorff by 2.16 (iii). So (X, τ 2 ), the top ological space induced fro m X R 2 via q : X → X R 2 will b e pre-Haus do rff. It is r eadily sho wn that the a ssignmen t (X, τ ) 7− → (X, τ 2 ) is left adjoint to the inclusio n T 0 , 2 - TO P ֒ → TOP . 3. Replacing Hausdorff With Pre-Hausdorff 8 There a re many known results in top ology which c o ncern Hausdorff spa ces. Given suc h a result, a natural questio n is whether o r not the result remains true when Hausdorff is replaced with pre-Hausdor ff. In this section we p oint out some sta ndard Theorems which can b e generalized to the pre-Hausdo rff s e tting, and so me which cannnot. Pre-Hausdo r ff top ologies share some inv ar iance pro perties with Hausdorff topo logies: PROPOSITION 3 .1. (i) Each su bsp ac e of a pr e-H ausdo rff sp ac e is also pr e- Hausdorff. (ii) The Cartesian pr o duct of pr e-Hausdorff sp ac es is also pr e-H ausdorff. Pr o of: T he pro of of (i) is straightforw ar d, (ii) follows immediately from Theorem 1.6. In the following result, as in [11] page 1 12, we make a distinction betw een normal spaces and T 4 spaces; namely , nor mal spaces need not hav e clos ed p oints. Recall that every compac t Hausdor ff space is nor ma l (see [11], p1 4 1). THEOREM 3.3. Every compact pre-Hausdor ff space is normal. Pr o of: Suppose (X , τ ) is a compa ct pr e-Hausdorff space. Since the Theorem is trivially true when τ is the indiscre te topo logy , we assume that it is no t and choose a clo sed set A ⊂ X and a point x / ∈ A. F or e ac h y ∈ A, A C provides a T 0 separation o f x and y . Since X is pre-Haus do rff, x and y have a T 2 separation b y , say N y ∋ y and N x,y ∋ x. Then { N y } y ∈ A is an ope n cov er o f A. Since A is compact, ∃ y 1 , ... , y n ∈ A such that U = n \ i =1 N x,y i and V = n [ i =1 are op en, and they pr ovide a disjoint separa tion of x and A. Now supp ose that A and B a re disjoint, closed sets in X. By the ab ov e we hav e, ∀ a ∈ A, ∃ U a , U a,B ∈ τ s uc h tha t a ∈ U a , B ⊂ U a,B , and U a ∩ U a,B = ∅ . Since A is compact, ∃ a 1 , ... , a n such that U A = n [ i =1 U a i and V B = n \ i =1 U a i ,B are b oth op en, and they pr ovide a disjoint separa tion of A a nd B. A sp e c ia l prop ert y of Haus do rff topolo gies is the following: Each finite subset of a Hausdo rff space is clo sed. This is clear ly not shared with pre-Hausdor ff top ologies; for if X is a finite indiscrete spa ce with more than one element, for instance, and A is a n y non-v oid prop er subset of X, then A is not closed. Many imp ortant r esults in volve mating co mpactness with the Hausdor ff prop ert y . An in triguing feature of compact Hausdorff spaces is that they are essentially alge braic. Indeed, a well-known result is that the catego ry of com- pact Hausdorff spaces is algebra ic (or ”monadic”) ov er the categor y of sets (see [12], chapter 6). A fac t which is crucial in proving this is the following: 9 If X is co mpact a nd Y is Hausdorff, then any contin uous function f : X → Y is a clo s ed map. This result is clearly fals e for pre-Hausdorff spaces ; for example, if we map a compact space (X, τ ) whic h is not indiscrete into X with the indiscrete top ology b y the identit y function, then we hav e a contin uous bijection which is not a closed map. Consequently , this identit y map is not a homeomorphism. It is easily shown that if a categ o ry A is a lgebraic o ver a category B (i.e., A is isomorphic to a categor y of T-a lgebras, where T is a mona d in X determined b y an adjunction), and L ⊣ R : A → B is the adjoint pair of functors which determines the isomo r phism, then A sa tisfies: if f : a → b is an y morphism in A, and R(f ) : R(a) → R(b) is an isomorphism in B, then f is an iso morphism in A. In the case of compact Hausdo rff spaces ov er the c a tegory of se ts, this rev e a ls the well-known fact that a contin uous bijection of compact Hausdorff spa ces is a ho meomorphism. Since the example in the preceeding paragr aph shows a contin uous bijection from a compact space to a pre-Hausdorff spa ce whic h is not a homeomorphism, we conclude tha t the ca tegory of co mpact pre-Hausdorff spaces is no t algebraic over the ca tegory of s ets. References 1. Andrews, G. E.: The The ory of Partitions, Cambridge Univ er s it y Press, 1998. 2. Arenas, F. G.: Alexandroff Spaces, A cta Math. Univ. Comenianae 68 (1999), 1 7-25. 3. Das, P . and B anerjee, A. K.: Pairwise Borel and Baire Measures in Bispaces, Ar c. Math. (Brno) 41 (2005), 5-1 5 . 4. Dugundji, J.: T op olo gy, Allyn and Bacon, 1966 . 5. F ar rag, A. S. and Abbas, S. E.: Computer Pro gramming for Construction and Enumeration of a ll Regular T o polog ies and Equiv alence Relations o n Finite Sets, Appl. Math. 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