Uniform covers at non-isolated points

UNIF ORM CO VERS A T NON-ISOLA TED POINTS FUCAI LIN AND SHOU LIN Abstract. I n this pap er, the authors define a space with an uniform base at non-isolated p oi n ts, give some c haracterizat ions of images of metric spaces by boundary-compact maps, and study certain relationship among spaces with special base prop erties. The main resul ts are the following: (1) X is an open, boundary-compact image of a metric space if and only if X has an uniform base at non-i solated points; (2) Eac h d iscretizable s pace of a space with an uniform base is an op en compact and at most b oundary-one image of a space with an unif orm base; (3) X has a p oin t-coun table ba se if and only if X i s a bi-quotien t, at most b oundary-one and coun table-to-one image of a metric space. 1. Intr oduction T op ologists obtained many interesting characteriza tions of the images of metric spaces by s ome kind of maps. A . V. Arhangel’ski ˇ ı [3] pro ved that a space X is an op en compact image o f a metric spac e if and only if X ha s an uniform base. Recently , C. Liu [16] gives a new characteriza tion of s paces with a point-coun table base b y pseudo-op en and a t mo st b oundary-one images of metr ic spa ces. Ho w to character an open or pseudo- open and bo undary-compact images of metric s pa ces? On th e other hand, a study of spaces with a sharp base o r a weakly uniform base [5 , 6] shows that some prope rties of a non-iso lated p oint set o f a top ological space will help us discuss a whole constructio n o f a space. In this pa per, the author s analyze some base pro perties o n non-isolated po in ts of a space, in tro duce a space having an uniform bas e at non-iso lated points and describe it as a n image of a metric spa ce by op en boundary - compact m aps. Some relationship among the images o f metr ic spaces under op en b oundary-compact maps, pseudo-op en b oundary-compa ct maps , op en compact maps, and space s with a p oint-coun table base ar e dis cussed. By R , N , denote the s e t of real n umbers a nd positive in tegers, r espectively . F or a s pace X , let I ( X ) = { x : x is an isola ted p oin t of X } and I ( X ) = {{ x } : x ∈ I ( X ) } . In this pap er all spaces a r e T 2 , all ma ps ar e contin uous a nd onto. Reca lled some basic definitions. Let X be a top ological space. X is called a metac omp act (resp. p ar a c omp act , meta-Lindel¨ of ) space if every o pen cover of X has a point-finite (resp. loc a lly finite, po in t-countable) open refinemen t. X is said t o ha ve a G δ - diagonal if the diagonal 2000 Mathematics Subje ct Classific ation. 54C10; 54D70; 54E30; 54E40. Key wor ds an d phr ases. Boundary-compact maps; developable spaces; un ifor m bases; sharp bases; open maps; pseudo-open maps. Supported in part by the NSFC( No. 10571151). 1 2 FUCAI LIN AND S HOU LIN ∆ = { ( x, x ) : x ∈ X } is a G δ -set in X × X . X is called a p erfe ct sp ac e if every op en subset o f X is an F σ -set in X . Definition 1.1. Let P b e a base of a spa ce X . (1) P is a n uniform b ase [1] (resp. uniform b ase at non-isolate d p oints ) for X if for eac h ( r esp. non-iso lated) point x ∈ X and P ′ is a coun tably infinite subset of ( P ) x , P ′ is a neighborho o d bas e at x . (2) P is a p oint-r e gular b ase [1] (resp. p oint-r e gular b ase at non- isola te d p oints ) for X if for each ( re sp. non-isolated) point x ∈ X and x ∈ U with U op en in X , { P ∈ ( P ) x : P 6⊂ U } is finite. In the definition, “at non- isolated points” means “at each non-isolated point of X ”. It is obvious tha t uniform bases (r e s p. point-regular ba ses) ⇒ unifor m bases at non-isolated p oints (res p. po in t-regula r bases a t non-isolated p oints), but we will see that uniform ba ses at non-isolated p oints (resp. po in t-regula r ba s es at non-isolated points) 6⇒ uniform bases (r esp. point-regular bas es) by Example 4.1. Definition 1.2. Let X b e a spac e , and {P n } a seq uence of open subsets of X . (1) { P n } is ca lled a quasi-development [8] for X if for every x ∈ U with U op en in X , ther e exis ts n ∈ N such that x ∈ st( x, P n ) ⊂ U . (2) { P n } is called a development (resp. development at non-isolate d p oints ) for X if { st( x , P n ) } n ∈ N is a neig h b orho od base a t x in X for ea c h (resp. non-isolated) po in t x ∈ X . (3) X is called quasi-develop able (resp. develop able , develop able at non-isolate d p oints ) if X has a quas i- dev elopment (res p. development, developmen t at non-isolated po in ts). It is obvious that every development for a space is a developmen t at non-iso lated po in ts, but a s pace having a dev elopment at non-isolated p oint s may not have a developmen t, see Example 4.2. Definition 1.3. Let f : X → Y be a ma p. (1) f is a c omp act map (resp. s-map ) if e ac h f − 1 ( y ) is co mpa ct (resp. separa - ble) in X ; (2) f is a b oundary-c omp act map (r esp. b oundary-finite map , at most b oundary- one map ) if ea c h ∂ f − 1 ( y ) is compact (resp. finite, at most one p oint) in X ; (3) f is a n op en map if whenever U op en in X , then f ( U ) is op en in Y ; (4) f is a bi-quotient map (resp. c ount ably bi -quotient map ) if for any y ∈ Y and any (resp. coun table) family U of op en subse ts in X with f − 1 ( y ) ⊂ ∪U , there exists finite subset U ′ ⊂ U such tha t y ∈ Int f ( ∪U ′ ); (5) f is a pseudo-op en map if whenever f − 1 ( y ) ⊂ U with U o p en in X , then y ∈ Int( f ( U )). It is easy to see that op en maps ⇒ bi-quotient maps ⇒ co untably bi-quotien t maps ⇒ pseudo- open ma ps ⇒ quotient maps. Definition 1.4. Let X b e a spac e . (1) A collection U o f subsets of X is s a id to be Q (i.e., interior-pr eserving ) if Int ( ∩W ) = ∩{ Int W : W ∈ W } for every W ⊂ U . UNIFORM COVERS A T NON-ISOLA TE D POINTS 3 (2) An ortho-b ase [17] B for X is a base of X such that either ∩ A is op en in X or ∩A = { x } / ∈ I ( X ) and A is a neigh b orho o d base at x in X fo r each A ⊂ B . A space X is a pr oto-metrizable sp ac e [13] if it is a paracompact space with an ortho -base. (3) A sharp b ase [2] B of X is a base of X suc h that, for every injective sequence { B n } ⊂ B , if x ∈ T n ∈ N B n , then { T i ≤ n B i } n ∈ N is a neigh bo rhoo d base at x . (4) A base B of X is said to b e B C O (i.e., bases of co untable orders) if, fo r any x ∈ X , { B i } ⊂ B is a strictly decreasing sequence, th en { B i } i ∈ N is a neighborho o d base at x . It is well known that [2, 5, 6] (1) Unifor m bases ⇒ σ -po in t-finite bases ⇒ σ -Q bases; (2) Unifor m bases ⇒ s ha rp bases , developable spaces ⇒ BCO, G δ -diagona ls ; (3) Sha r p bases ⇒ point-count able bases. Readers may refer to [11, 18] for unstated definitions and ter minology . 2. Some lemmas In this sec tio n some tec hnical lemmas are given. Lemma 2.1. L et P b e a b ase for a sp ac e X . Then the f ol lowing ar e e quivalent: (1) P is an u niform b ase at non-isolate d p oints for X ; (2) P is a p oint-r e gular b ase at non-isol ate d p oints for X . Pr o of. (2) ⇒ (1) is trivial. W e only need to prove (1) ⇒ (2). Let P b e an uniform base at non- isolated points for X . If there exist a non- isolated p oin t x ∈ X and an op en subset U in X with x ∈ U s uc h that { P ∈ ( P ) x : P 6⊂ U } is infinite. T ake { P n : n ∈ N } ⊂ { P ∈ ( P ) x : P 6⊂ U } , and choos e x n ∈ P n \ U for eac h n ∈ N . Then { P n } n ∈ N is a neighbor hoo d base at x , th us t he sequence { x n } conv e rges to x in X . Hence x m ∈ U f or some m ∈ N , a con tradiction. Therefore, P is a p oint -re g ular base a t non-is olated p oin ts for X .  Lemma 2. 2. L et {P n } b e a develop ment at non-isola te d p oints for a sp ac e X . If P n is p oint-fin ite at e ach non-isolate d p oint and P n +1 r efin es P n for e ach n ∈ N , then P = I ( X ) ∪ ( S n ∈ N P n ) is an uniform b ase a t non-isolate d p oints for X . Pr o of. Let x b e a non-isolated point in X , and { P m : m ∈ N } an infinite subset of ( P ) x . By the p oin t-finiteness, there exists P m k ∈ P n k such that m k < m k +1 , and n k < n k +1 for each k ∈ N . Since {P n } is a development a t non-isolated po in ts for X , { P m k } k ∈ N is a neig h b orho od ba se at x in X , so { P m } m ∈ N is a neig h b orho od base at x . Th us P is a unifor m bas e at non-isolated points for X .  Let P b e a family of subsets of a space X . P is called p oint-finite at non-isolate d p oints (resp. p oint-c ountable at non-isolate d p oints ) if f or each non-is o lated p oin t x ∈ X , x is b elong to at most finite(resp. countable) elements of P . Let {P n } b e a developmen t (r esp. a de velopment at no n-isolated p oints) for X . {P n } is said to b e a p oint-finite deve lopment (resp. a p oint-finite d evelopment a t non-isolate d p oints ) for X if eac h P n is point-finite at each (resp. non-isola ted) p oin t of X . Lemma 2.3. A sp ac e X has an uniform b ase a t non-isolate d p oints if a nd only if X has a p oint- finite development at n on-isolate d p oints. 4 FUCAI LIN AND S HOU LIN Pr o of. Sufficiency . It is easy to see b y Lemma 2.2 . Necessity . Let P be an uniform ba se at no n-isolated p oin ts for X . Then P is a po in t-regula r ba s e at non-isolated p oin ts b y Lemma 2.1. W e can ass ume that if P ∈ P and P ⊂ I ( X ), P is a single point set. Claim: Let x b e a non-iso lated p oin t of X and y 6 = x . Then { H ∈ P : { x, y } ⊂ H } is finite. In fact, { H ∈ P : { x, y } ⊂ H } ⊂ ( P ) x . If { H ∈ P : { x, y } ⊂ H } is infinite, then it is a loca l base at x , hence y → x , this is a contradiction. (a) P is po in t-countable at non-isolated p oints in X . Let x ∈ X b e a non-iso lated p oint, there is a no n-trivial sequence { x n } converging to x . By the Cla im, { P ∈ ( P ) x : x n ∈ P } is finite for e a c h n , then ( P ) x = S n ∈ N { P ∈ ( P ) x : x n ∈ P } is coun table. A family F of subsets of X is called having the prop ert y ( ♯ ) if for any F ∈ F \ I ( X ), then { H ∈ F : F ⊂ H } is finite. (b) P has the proper t y ( ♯ ). Since F ∈ P \ I ( X ). Then F co n tains a non-isolated point and | F | > 1. By the Claim, P has the prop erty ( ♯ ). Put P m = { H ∈ P : if H ⊂ P ∈ P , then P = H } ∪ I ( X ), P ′ = ( P \ P m ) ∪ I ( X ) . (c) P m is a n op en c over, and is p oint-finite at non-isolated points for X . There exists H P ∈ P m such that P ⊂ H P for each P ∈ P \ I ( X ) by (b). Thus P m is an op en co ver of X . If P m is not po in t-finite at some non-isola ted p oint x ∈ X , then there exists an infinite subset { H n : n ∈ N } of ( P m ) x . F or ea c h n ∈ N , H n 6⊂ H 1 , there exists x n ∈ H n +1 \ H 1 . Then x n → x ∈ H 1 , a contradiction. (d) P ′ is a p oint-regular base at no n-isolated p oint s for X . Let x ∈ U \ I ( X ) with U o p en in X . There exist V , W ∈ P and y ∈ V \ { x } such that x ∈ W ⊂ V \ { y } ⊂ V ⊂ U . Th us W ∈ P ′ . Then P ′ is a base for X , and it is a p oint-regular base at non-isolated p oints for X . Put P 1 = P m and P n +1 = [( P \ S i ≤ n P i ) S I ( X )] m for any n ∈ N . Then P = S n ∈ N P n by (b). (e) {P n } is a po in t-finite dev elopment at non-isolated p oints for X . Each P n is point-finite at non-isolated points b y (c) and (d). If x ∈ U \ I ( X ) with U op en in X , then { P ∈ ( P ) x : P 6⊂ U } is finite, thus there is n ∈ N such that P ⊂ U whenever x ∈ P ∈ P n , i.e., st( x, P n ) ⊂ U . So { P n } is a development at no n- isolated p oin ts.  Lemma 2.4. [3, 4, 14] The fol lowing ar e e quivalent fo r a s p ac e X : (1) X is a n op en c omp act image of a met ric sp ac e; (2) X is a n pse udo-op en c omp act image of a metric sp ac e; (3) X has an uniform b ase; (4) X has a p oint-r e gular b ase; (5) X is a metac omp act and develop able sp ac e; (6) X is a sp ac e with a p oint- finite development. Lemma 2.5. Each pseudo-op en, b oundary-c omp act map is a bi-quotient map. Pr o of. Let f : X → Y b e a pseudo-op en, boundar y-compact ma p. F or each y ∈ Y and a family U of op en subsets in X with f − 1 ( y ) ⊂ ∪U , ∂ f − 1 ( y ) ⊂ ∪U ′ for some finite U ′ ⊂ U . W e ca n assume that there exists U ∈ U ′ such that U ∩ f − 1 ( y ) 6 = ∅ . UNIFORM COVERS A T NON-ISOLA TE D POINTS 5 Thu s y ∈ f ( U ). Le t V = ( ∪U ′ ) ∪ Int( f − 1 ( y )). Then f − 1 ( y ) ⊂ V . Since f is pseudo- op en, thus y ∈ Int( f ( V )) ⊂ f (( ∪U ′ ) ∪ f − 1 ( y )) = f ( ∪U ′ ) ∪ { y } = f ( ∪U ′ ) , so f ( ∪U ′ ) is a neighborho o d of y in Y . Hence f is a bi-quotient map.  3. Main Resul ts In this section s paces with an uniform base a t non-isolated p oints a re discus sed, and some c har a cterizations o f images o f metric spaces by b oundary-compact maps are g iven. Theorem 3.1. The f ol lowing ar e e quivalent for a sp ac e X : (1) X is a n op en, b oundary-c omp act image of a metric sp ac e; (2) X has an uniform b ase at non- isola te d p oints; (3) X has a p oint-r e gular b ase at non- isola te d p oints; (4) X has a p oint-finite development at non-isolate d p oints. Pr o of. It is obvious that (2) ⇔ (3) ⇔ (4) b y Lemma 2.1 and Lemma 2 .3. (1) ⇒ (4). Let M be a metric space and f : M → X an op en, boundar y-compact map. By [11, 5.4.E], we can ch o ose a sequence {B i } o f op en cov ers of M such that { st( K, B i ) } i ∈ N is a neighborho o d base of K in M for e a c h co mpact subset K ⊂ M . F or e a c h i ∈ N , we can assume t hat B i +1 is a locally finite open refinemen t of B i , and set P i = f ( B i ) ∪ I ( X ). Then P i is an open cover of X for each i ∈ N . If x is an accumulation p oint of X , then Int f − 1 ( x ) = ∅ , thus f − 1 ( y ) = ∂ f − 1 ( x ) is compact in M , hence { B ∈ B i : B ∩ f − 1 ( x ) 6 = ∅ } is finite b y the lo cal finiteness of B i , i.e., ( P i ) x is finite. This shows that P i is p oin t-finite at non- is olated p oints. Next, we will prov e that {P i } is a developmen t at non-is o lated p oints for X . Let x ∈ U \ I ( X ) with U o pen in X . Since f − 1 ( x ) is compact, there exists m ∈ N such that st( f − 1 ( x ) , B m ) ⊂ f − 1 ( U ), so st( x, P m ) = st( x, f ( B m )) ⊂ U . Thus {P i } is a po in t-finite dev elopment at non-isolated p oints for X . (4) ⇒ (1). Firs t, a metric space M and a function f : M → X are defined as follows. Let { P n } b e a point-finite development at non-iso lated points for X . F or each n ∈ N , ass ume that I ( X ) ⊂ P n , put P n = { P α : α ∈ Λ n } and endo w Λ n a discrete topolo gy . Put M = { α = ( α n ) ∈ Y n ∈ N Λ n : { P α n } n ∈ N is a neighbo rhoo d bas e at some x α ∈ X } . Then M , whic h is a s ubspace o f the pro duct space Q n ∈ N Λ n , is a metric space. Define a function f : M → X b y f (( α n )) = x α . Then f (( α n )) = T n ∈ N P α n , and f is w ell defined. ( f , M , X, P n ) is called a Ponomarev system. It is easy to see that f is a map. The following will pro ve that f is an op en b oundary- compact map. (a) f is an op en map. F or any α = ( α n ) ∈ M , n ∈ N , put B ( α 1 , α 2 , · · · , α n ) = { ( β i ) ∈ M : β i = α i whenever i ≤ n } . 6 FUCAI LIN AND S HOU LIN Then f ( B ( α 1 , α 2 , · · · , α n )) = T i ≤ n P α i . In fact, if β = ( β i ) ∈ B ( α 1 , α 2 , · · · , α n ), f ( β ) = T i ∈ N P β i ⊂ T i ≤ n P α i . Th us f ( B ( α 1 , α 2 , · · · , α n )) ⊂ \ i ≤ n P α i . On the other hand, let x ∈ T i ≤ n P α i . Cho ose a countable family { P β i } i ∈ N of subsets of X suc h tha t (i) x ∈ P β i ∈ P i for eac h i ∈ N , (ii) β i = α i whenever i ≤ n , and (iii) P β i = { x } whenev er i > n and x ∈ I ( X ). Put β = ( β i ). Then β ∈ B ( α 1 , α 2 , · · · , α n ), and f ( β ) = x . Thus T i ≤ n P α i ⊂ f ( B ( α 1 , α 2 , · · · , α n )). In conclusion, f ( B ( α 1 , α 2 , · · · , α n )) = T i ≤ n P α i . Since { B ( α 1 , α 2 , · · · , α n ) : ( α i ) ∈ M , n ∈ N } is a base of M , f is an op en map. (b) f is a b oundary-compac t map. Let x ∈ X . If x ∈ I ( X ), then ∂ f − 1 ( x ) = ∅ . If x 6∈ I ( X ), ∂ f − 1 ( x ) = f − 1 ( x ) by (b). F o r each i ∈ N , let Γ i = { α ∈ Λ i : x ∈ P α } . Then Γ i is finite. Thus Q i ∈ N Γ i is a compact subset of Q i ∈ N Λ i . W e only need to pro of f − 1 ( x ) = Q i ∈ N Γ i . Indeed, if α = ( α i ) ∈ Q i ∈ N Γ i , then { P α i } i ∈ N is a neig h b o rho od bas e at x fo r X . Thus α ∈ M and f ( α ) = x , so Q i ∈ N Γ i ⊂ f − 1 ( x ). O n the other ha nd, if α = ( α i ) ∈ f − 1 ( x ), then x ∈ T i ∈ N P α i and α ∈ Q i ∈ N Γ i . So f − 1 ( x ) ⊂ Q i ∈ N Γ i . Thu s ∂ f − 1 ( x ) = f − 1 ( x ) = Q i ∈ N Γ i is compact.  In the Ponomarev sy stem ( f , M , X , P n ), it is a lways hold that f − 1 ( x ) ⊂ Q i ∈ N { α ∈ Λ i : x ∈ P α } fo r each x ∈ X . The following cor ollary is obtained. Corollary 3.2. A sp ac e X has a p oint-c ountable b ase which is unifo rm at non- isolate d p oints if and only if X is an op en b oundary-c omp act, s -image of a metric sp ac e. Corollary 3.3. Each sp ac e having an uniform b ase at non-isolate d p oints is pr e- serve d by an op en, b oundary-finite map. Pr o of. let f : X → Y b e an o pen b o undary-finite map, where X has an uniform base at non-iso lated po ints. There exist a metric space M and an open b oundary- compact map g : M → X b y Theor em 3.1. Since ∂ ( f ◦ g ) − 1 ( y ) ⊂ S { ∂ g − 1 ( x ) : x ∈ ∂ f − 1 ( y ) } for each y ∈ Y , f ◦ g : M → Y is an op en b oundary-compact ma p. Hence Y has an unifor m base at no n-isolated p oints.  Theorem 3. 4. L et X b e a sp ac e having an u niform b ase at non- isolate d p oints. Then (1) X is a quasi-develop able sp ac e; (2) X has an ortho-b ase and a σ -Q b ase. Pr o of. By Theor e m 3 .1, let {P n } n ∈ N be a point-finite developmen t at non-isolated po in ts for X . Put P 0 = I ( X ). It is easy to c heck that { P n } n ∈ ω is a quasi- developmen t for X . Let P = S n ∈ ω P n . Then P is a σ -Q base and an ortho - base fo r X . UNIFORM COVERS A T NON-ISOLA TE D POINTS 7 First, P n is in terior -preserving for eac h n ∈ N . Indeed, for each A ⊂ P n , if x ∈ ∩A − I ( X ), then ( P n ) x is finite, th us ∩A is a neighbo rho o d of x in X . So P is a σ -Q base fo r X . Secondly , let A ⊂ P with ∩A not op en in X . Then there exists x ∈ ∩A suc h that ∩A is not a neighbo rhoo d of x in X , thus x is a non-isolated p oin t and ( P n ) x is finite for eac h n ∈ N . Let x ∈ U with U op en in X . There exists n ∈ N such that x ∈ st( x, P n ) ⊂ U . Cho ose m ≥ n and A ∈ A ∩ P m . Then A ⊂ st( x, P n ) ⊂ U , th us A is a neighbo r hoo d base a t x in X . So ∩A is a single p oint subset. Hence P is a n ortho-ba se for X .  Corollary 3. 5. L et X b e a sp ac e having an u n iform b ase at non-isolate d p oints. Then (1) ⇒ (2) ⇔ (3) in the fol lowing: (1) X has a sharp b ase; (2) X is a develop able sp ac e; (3) I ( X ) i s a n F σ -set in X . Pr o of. (1) ⇒ (3) is proved in [7 , Theorem 3.1 ] for any space X . (2) ⇒ (3) is o b vious bec ause each op en s ubset of a dev elopa ble s pace is an F σ -set. (3) ⇒ (2). Let {B n } be a point-finite development at non-isolated p oints for X b y Theorem 3.1. Since I ( X ) is an F σ -set, there exists a sequence { G n } of open subsets o f X such that X − I ( X ) = T n ∈ N G n . F or each n ∈ N , let U n = { G n } ∪ {{ x } ; x ∈ X − G n } . Then {B n , U n } is a developmen t for X . Hence X is a developable space.  The following cor ollary is hold by Lemma 2.4. Corollary 3.6. A sp ac e X is an op en c omp act image of a metric sp ac e if and only if X is a p erfe ct, metac omp act sp ac e, whi ch is an op en b oundary-c omp act image of a metric sp ac e. By the corollar y , some metrizable theor ems on spac e s with an unifor m base at non-isola ted p oints can b e obtained. F or exa mple, let X b e a space with an uniform bas e at non- isolated p oints, then X is metriza ble if and only if it is a per fect, co lle ction wise nor mal space. Now, a s pecial space with an uniform base at non-isolated points is discussed. Let ( X, τ ) b e a s pa ce a nd A ⊂ X . X is s aid to b e discr etizable b y A if X is endow ed with the topolo gy generated by τ ∪ {{ x } : x ∈ A } as a base for X [1 7]. Denote the discretizable s pace of X by X A . It is ob vious that the top ology o f a space X is coarser tha n the discretizable top ology of X A . If X has an uniform base, then X A not only has a G δ -diagona l and an un iform base at non-is olated po in ts, but also has a σ - point finite base. In [13, Theore m 3 .1] ha s shown that a space is a discr etization of a metric space if and only if it is a pro to-metrizable space ha ving a G δ -diagona l. Theorem 3.7. Each discr etizable sp ac e of a sp ac e ha ving an u niform b ase is a n op en c omp act and at most b oundary-one image of a sp ac e ha ving an uniform b ase. Pr o of. Let X be a space having an unifor m base. By Lemma 2 .4 , there is a po in t- finite developmen t {U m } for X , where U m +1 refines U m for each m ∈ N . F or each A ⊂ X , put H = ( X × { 0 } ) ∪ ( A × N ); 8 FUCAI LIN AND S HOU LIN V ( x, m ) = { x } × ( { 0 } ∪ { n ∈ N : n ≥ m } ) , x ∈ X , m ∈ N ; W ( J, m ) = (( J ∩ ( X − A )) × { 0 } ) ∪ (( J ∩ A ) × { n ∈ N : n ≥ m } ) , J ⊂ X, m ∈ N . Endow H with a base consisting of the following elements: V ( x, m ) , ∀ x ∈ A, m ∈ N ; W ( J, m ) , ∀ op e n subset J ⊂ X , m ∈ N ; { x } , x ∈ A × N . Then H is a T 2 -space. F or any m ∈ N , let P m = { V ( x, m ) : x ∈ A } ∪ { W ( U, m ) : U ∈ U m } ∪{{ h } : h ∈ A × { 1 , 2 , · · · , m − 1 }} . Then {P m } m ≥ 2 is a po in t-finite developmen t for H . Hence H has an uniform base. Let π 1 | H : H → X A be the pro jective map. It is easy to s ee that π 1 | H is an op en compact and at most b oundary -one map.  Hence, each dis cretizable space of a spa c e having an unifor m base is in MOBI [8]. C. Liu [16] g a ve some c hara cterizations of quotient (resp. pseudo -open) bo undary- compact images of metric s paces. The following ar e further results. Theorem 3.8. The f ol lowing ar e e quivalent for a sp ac e X : (1) X is first-c ountable; (2) X i s an image of a metric sp ac e under a pseudo-op en, at most b oun dary-o ne (r esp. b oundary-c omp act) ma p; (3) X is an image of a metric sp ac e under a bi-quotient, at m ost b oundary-one (r esp. b oundary-c omp act) ma p. Pr o of. (1) ⇔ (2) was prov ed in [16, Corollar y 2.1], and (2) ⇔ (3) is true by Lemma 2.5.  Theorem 3.9. The f ol lowing ar e e quivalent for a sp ac e X : (1) X has a p oint-c ountable b ase; (2) X is a c ount ably bi-quotient, s -image of a metric sp ac e; (3) X is a pseudo-op en, b oun dary-c omp act and s -image of a metric sp ac e; (4) X is a bi-quotient, at most b oundary-one and c ountable-to-one image of a metric sp ac e. Pr o of. C. Liu proved that a space has a p oin t-countable base if and only if it is a pseudo-op en, a t most b oundary- one and co untable-to-one image of a metric s pace in [16]. Thus (1) ⇔ (4) by Lemma 2.5. (4) ⇒ (3) is trivial. (3) ⇒ (2) b y Lemma 2.5, and (2) ⇔ (1) by [21].  4. Examples In this section some examples ar e giv en, which sho w certain relations among bo undary-compact ima g es of metric spac es and generalized metric spac e s. Example 4.1 . Let X b e the closed unit int erv al I = [0 , 1] and B a Berns tein subset of X . In other w ords, B is an unco un table set whic h contains no uncountable clo s ed subset o f X . The discretizable space X B is ca lled Michael line [20]. UNIFORM COVERS A T NON-ISOLA TE D POINTS 9 Let X ∗ be a copy of X B , a nd f : X B → X ∗ a homeo morphism. Put Z = X B L X ∗ , and let Y a quotient s pa ce obtained from Z b y identif ying { x, f ( x ) } to a p oint for each x ∈ X B \ B . Then (1) X B is a discretizable space of the metric space I , so it is a pro to-metrizable space, a nd an o pen compact, at most bo unda ry-one image of a space with an uniform ba se by Theorem 3.7. (2) X B is not a BCO space, hence it is not an op en compact image of a metric space; (3) Y is an op en bo undary-compact, s -ima ge of a metric space; (4) Y has not any G δ -diagona l b y [23, Ex a mple 1]. It is ob vious that X B is a paracompact space which is a discretizable space of the metric space I . If X B is BCO, it is a dev elopable space, then B is an F σ -set in X B , a contradiction. Th us X B is no t BCO. It is easy to c heck that Y has a point-coun table base whic h is uniform at non- isolated points. Hence Y is an op en b oundary- compact, s -ima g e of a metric space by Corollar y 3.2 . Example 4.2 . Let ψ ( D ) b e Isb el l-Mr´ owka space [22], here | D | ≥ ℵ 0 . Then (1) ψ ( D ) is an op en, bo unda ry-compact imag e of a metric space; (2) ψ ( D ) is not a meta-Lindel¨ of space; (3) ψ ( D ) is a developable spac e if | D | = ℵ 0 ; (4) ψ ( D ) is not a p erfect space if | D | ≥ c . A co llection C of subsets o f an infinite set D is said to b e almost disjoint if A ∩ B is finite whenever A 6 = B ∈ C . Let A b e an almost disjoint c o llection of coun tably infinite subsets of D and maxima l with res p ect to the pr o perties . Then |A| ≥ | D | + [15]. Isbell-Mr´ owk a space ψ ( D ) is the set A ∪ D endow ed with a top ology as follo ws: The p oin ts of D are isola ted. Basic neighbo r hoo ds of a p oint A ∈ A ar e the sets of the form { A } ∪ ( A − F ) where F is a finite subset o f D . Let X = ψ ( D ) , A = { A α } α ∈ Λ and each A α = { x ( α, n ) : n ∈ N } . F or ea c h n ∈ N , put B n = {{ A α } ∪ { x ( α, m ) : m ≥ n } : α ∈ Λ } ∪ {{ x } : x ∈ D } . It is easy to see that {B n } is a point-finite developmen t for X . Th us X is the op en, b oundary-co mpact ima ge of a metric spac e by Theorem 3.1. Since an op e n cov er {{ A α } ∪ D } α ∈ Λ of X ha s not any p oint-coun table op en r efinemen t, X is not a meta-Lindel¨ of spa c e. Thus X is no t a n o pen s -image of a metric space, a nd X is not a discretizable space of a space with a n uniform base b y Theo rem 3.7 . If D is coun table, it is ob vious t hat ψ ( D ) is a developable space. Hence ψ ( D ) has a G δ -diagona l, but ψ ( D ) has no t any p oint-coun table base b ecause ψ ( D ) is not a meta -Lindel¨ of space. If | D | ≥ c , ψ ( D ) is not a dev elopable space [9], thus ψ ( D ) is not p erfect by Corollar y 3.5 . Example 4.3 . Ther e is a space X such that (1) X has a sha r p base; (2) X has not an uniform base at non-iso lated po in ts; (3) X is an open compact and countable-to-one imag e o f a space with an uniform base. A space X having the prop erties is construc ted in [2, Example 5.1 ], where it is shown that X has a no n-dev elopa ble space with a shar p base. Since X has not any 10 FUCAI LIN AND S HOU LIN isolated p o in t, it is not an op en, bo undary-compact image of a metric space and has not an uniform base at non-iso lated p oint s by Theorem 3.1. J. Chab er in [10, Example 4.5] was prov ed that X is an op en compact and coun table-to-o ne image of a space with a n uniform base. Example 4.4 . There is a bi-quo tien t, at most boundar y-one image X o f a metric space such that X is neither a ps eudo-op en s -image of a metric space, nor an op en, bo undary-compact ima g e of a metric space. Let X = R 2 endow ed wit h the but terfly top ology[19]. It is easy to see that X is a first-co un table, pa racompact s pace witho ut any isolated po in t. Since X is a first-countable space, then X is a bi-quotient, at most bounda ry-one imag e of a metric space by Theorem 3.8. Since X has not a p oint-coun table base [18, Example 1.8.3], X is not a countably bi-quotient s -image of a metric space by Theor em 3.9. Be cause each pseudo-op en map from a space onto a first-countable space is countably bi-q uotien t [21], X is not a pseudo- open s -image of a metric space. If X is an op en, b oundary-c ompact imag e of a metric space, X is an op e n compact imag e of a metric space for X do es not contain any iso lated p oint . So X is a dev elopable space by L e mma 2.4. Th us X is a metric space, a contradiction. Example 4.5 . There a proto-metrizable space w itho ut a n y uniform base at non- isolated p oints. G. Gruenhage in [12, p. 3 63] constr ucted a proto-metriza ble X which is not a γ - space. Hence X has not an y σ -Q base b y [18, P ropo sition 1 .7.10], and it has not any unifor m base at no n-isolated p oints by Theor e m 3 .4. Example 4.6 . There a space suc h that it is an open compact image of a metric space, whic h is not any op en, at most bounda ry-one image of a metric space. Y. T anak a in [24, Example 3 .1 ] constructed a no n-regular T 2 -space X which is an op en, at most tw o-to-o ne image of a metr ic s pace. S ince X has not any isolated p o in t, it is not an op en, at most b oundary- one image of a metric space. Otherwise, X is an imag e of a metr ic s pace under an op en and bijective map, then X is homeomorphic to a metric space, a contradiction. 5. Questions Some questions are p osed in the final. Question 5.1. L et a sp ac e X have a p oint-c ou n table b ase. If X has an uniform b ase at non-isolate d p oints, is X an op en , b oundary-c omp act, s -image of a metric sp ac e? Question 5.2. Is an op en and b oundary-c omp act s -image of a m et ric sp ac e an op en, b oundary-c omp act and c ount able-to-one image of a metric sp ac e? Question 5.3. How to char acter a discr etizable sp ac e of a sp ac e with an uniform b ase by a c ertain image of a metric sp ac e? F or example, whether the op en c omp act and at most b ou n dary-one image of a sp ac e with an u niform b ase is a discr etizable sp ac e of a sp ac e with an unifo rm b ase? Question 5.4. How to char acter a sp ac e which is an op en, at most b oundary-one, s -image of a metric sp ac e? UNIFORM COVERS A T NON-ISOLA TE D POINTS 11 Ac kno wledgeme n ts . The authors w ould like to thank the referee for his v a lu- able s uggestions. References [1] P . S. Aleksandro v, On the metrisation of topological spaces (in Russian), Bul l. A c ad. Po l. Sci., S´ er. Sci. Math. Astr on. Phys. , 8 (1960) , 135–140. [2] B. Al lec he, A. V. Arhangel’ski ˇ ı, J. Calbrix, W eak dev elopments and metrization, T op olo gy Appl. , 100 (2000), 23–38. [3] A. V. Arhangel’ski ˇ ı, On mappings of metric spaces (in Russian), Dokl. Akad. Nauk SSSR , 145 (1962), 245–247. [4] A. V. Arhangel’ski ˇ ı, Intersect ion of topologies, and pseudo-open bi compac t mappings (in Russian), Dokl. Akad. Nauk SSSR , 2 26 (1976) , 745–748. [5] A. V. Arhangel’ski ˇ ı, W. Just, E. A. Renziczenk o, P . J. Szept ycki, Sharp bases and weakly unifrom bases ve rsus p oi n t-coun table bases, T op olo gy Appl. , 100 (2000), 39–46. [6] C. E. Aull, A survey paper on some base axioms, T op olo gy Pr o c , 3 (1978), 1–36. [7] Z. Balogh, D. K. Burke , Two results on spaces with a sharp base, T op olo gy Appl. , 154 (2007), 1281–128 5. [8] H. R . Bennett, On Arhangel’ski ˇ ı’s class MOBI, Pr o c. Amer. Math. So c. , 26 (1970) , 178–180. [9] J. Chaber, Pr imitive generalizations of σ -spaces, T op olo gy. Col lo g. Math. So c. J´ anos Bolyai , 23 (1978). North-Holl and, Am sterdam, 1980 , 259–268. [10] J. Chaber, Mor e nondevelop able sp ac es in MOBI , Pro c. Amer. Math. So c., 103 (1988), 307– 313. [11] R. Engelking, Gener al T op olo gy (r evise d and c omplete d e dition) , Heldermann V erlag, Berlin, 1989. [12] G. Gruenhage, A note on q uasi-met rizability , Canad. J. Math., 29 (1977), 360–366 . [13] G. Gruenhage, P . Zenor, Pr oto-metrizable sp ac es , Houston J. Math., 3 (1977), 47–53. [14] R. W. Heath, Scr e enability, p ointwise p ar ac omp act ness and metrization of Mo or e sp ac es , Canad. J. Math., 16 (1964), 763–770. [15] K. Kunen, Set The ory: An Intr o duction to Indep endenc e Pr o ofs , North-Holland, Amsterdam, 1980. [16] C. Li u, A note on p oint-co untable w e ak b ase , Questions and Answ ers in General T op ology , 25 (2007), 57–61. [17] W. F. Lindg ren, P . J. Nyik os, Sp ac es with b ases satisfying c ertain or der and i nterse cti on pr op erties , P acific J. Math., 66 (1976), 455–476. [18] S. Lin, Gener alize d Metric Sp ac es and Mappings (in Chinese) , Chinese Science Press, Beijing, 1995. [19] L. F. McAuley , A r elation b etwe en p erfect se p ar ability, co mpleteness, and normality in semi- metric sp ac es , P acific J. Math., 6 (1956), 315–326. [20] E. A . Mic hael, The pr o duct of a normal sp ac e and a met ric sp ac e ne e d not b e normal , Bull. Amer. Math. So c., 69 (1963), 375–376. [21] E. A . Mic hael, A quintuple quotient quest , General T op ology Appl, 2 (1972), 91–138. [22] S. G. M r´ owk a, On c ompletely r e gular sp ac es , F und. Math., 72 (1965) , 998–1001. [23] V. Popov, A p erfect map ne ed not pr eserve d a G δ -diagonal , General T op ology Appl., 7 (1977), 31–33. [24] Y. T anak a, On op en finite-to-one maps , Bull. T okyo Gakugei Univ. IV, 25 (1973 ), 1–13. 12 FUCAI LIN AND S HOU LIN Fucai Lin: Dep ar tment of Ma themat ics, Zhang zhou Normal University, Zhangzhou 363000, P. R. China E-mail addr ess : lfc1979100 1@163.com Shou Lin (the corresponding author): Dep ar tment of Ma thema tics, Zhang zhou Nor- mal Un iversity, Zhang zhou 363000 , P. R. China; Institute of Ma thema tics, Ningde Teach- ers’ College, Ning de, Fujian 352100 , P. R. China E-mail addr ess : linshou@pu blic.ndptt.fj.cn