Regular Bases At Non-isolated Points And Metrization Theorems
In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space $X$ is a metrizable space, if and only if $X$ is a regular space with a $\sigma$-locally finite base at…
Authors: Fucai Lin, Shou Lin, Heikki Junnila
REGULAR BASES A T NON-ISOL A TED POINTS AND METRIZA TION THEOREMS FUCAI LIN, SHOU LIN, AND HEIKK I J. K. JUNNILA Abstra ct. In this pap er, w e define the spaces with a regular base at non-isolated p oints and discuss some metrization theorems. W e firstly show that a space X is a metrizable space, if and only if X is a regular space with a σ -lo cally finite base at non- isolated points, if and only if X is a perfect space with a regular base at n on-isolated p oints, if and only if X is a β -space with a regular base at non-isolated p oints. In addition, w e also discuss the relations b etw een the spaces with a regular base at non- isolated points and some generalized metrizable spaces. Finally , we giv e an affirmative answ er for a question p osed by F. C. L in and S. L in in [7], whic h also shows that a space with a regular base at non-isolated p oints has a p oint-coun table base. 1. Introduction The bases of top ological s paces o ccupy a core p osition in the stud y of the top ological theories and metrization problems, which has pro du ced many kinds of metrization the- orems, and establishes a foun dation f or the top ological dev elo pment [12]. F or example, the follo w ing is a classic metrizatio n theorem. Theorem 1.1. The fol lowing ar e e quiv alent for a sp ac e X : (1) X is metrizable; (2) X is a T 1 -sp ac e with a r e gu lar b ase; (3) X is a r e gu lar sp ac e with a σ -lo c al ly finite b ase. In recen t years, the theory of regular bases in top ological s p aces p lay ed an imp ortan t role in generalized metrizable spaces [2, 17]. On the other h and, in th e study of the theories of top ological spaces, w e are mainly concerned with the pr op erties of neigh b or- ho o ds on non-isolated p oin ts, and also discuss the relation b etw een their prop erties and global prop erties. F or example, a study of spaces with a s h arp base, a weakly uniform base or an uniform b ase at n on-isolated p oin ts [2, 3 , 7] sho ws that some prop erties of a non-isolated p oint set of a top ologica l sp ace will help us discuss the global construction of a spac e. Esp ecially , a space X w ith a uniform b ase at n on -isolated p oin ts if and only if X is the op en and b ou n dary-compact image of a metric space [7]. Th e most t ypical example is the spaces ob tained from a metrizable space by isolating the p oints of a su bset. Let B b e a base for a sp ace X . F or an y x ∈ X , the base B of X is calle d r e gular at a p oin t x if, for every n eigh b orho o d U of x , ther e exists an op en su bset V suc h th at x ∈ V ⊂ U and { B ∈ B : B ∩ V 6 = ∅ and B 6⊂ U } is fi nite. 2000 M athematics Subje ct Classific ation. 54D70; 54E35; 54D20. Key wor ds and phr ase s. Metrization; regular bases; locally-finite families; β -spaces; proto-metrizable; discretization; p oint-regular bases. Supp orted by the NSFC (No. 1097118 5) and the Educational Department of F ujian Pro vince(No. JA09166). 1 2 FUCAI LIN, SHOU LIN, AND HEIKKI J. K. JUN NILA By Th eorem 1.1, eve ry metric s pace has a base whic h is regular at n on -isolated p oin ts. Ho w ever, there exists a n on-metrizable sp ace with a b ase wh ic h is regular at non-isolated p oints, see the follo w in g Example 1.2. Example 1.2 . Let X b e the closed un it inte rv al I = [0 , 1] and B a Bernstein sub set of I . In other wo rds, B is an uncountable set w hic h cont ains no uncounta ble closed su bset of I . Endow X with th e follo wing top ology , i.e., Mic hael line [15]: G is an op en su bset for X if and on ly if G = U ∪ Z , where U is an op en subset of I with Euclidean top ology and Z ⊂ B . Let B b e a base of I with the Euclidean top ology , where B is regular at ev ery p oint of I . Then P = B ∪ {{ x } : x ∈ B } is a base for X and also regular at n on-isolated p oints. Hence this causes our in terests in a study of spaces with a base whic h is regular at non-isolated p oints, and the r elated p roblems of the m etrizabilit y . In this pap er , w e shall pro v e that spaces with a r egular base at n on-isolated p oint s are str ictly b et w een the discretizations of metrizable spaces and pr oto-metrizable s p aces, and w e also obtain some metrization theorems whic h h elp us to b etter un derstand the r elation b et w een the p rop erties at non-isolated p oint s and global p rop erties in the stu dy the generalized metrizable spaces. In this pap er all spaces are T 1 unless it is explicitly stated wh ic h separation axiom is a ssumed, and all maps are contin u ous and o n to. By R , N , denote the set of real n um b ers and p ositiv e in tegers, resp ectiv ely . F or a sp ace X , let I = I ( X ) = { x : x is an isolated p oin t of X } and I ( X ) = {{ x } : x ∈ I ( X ) } . Let P b e a family of subsets for X , and we denote st( x, P ) = ∪ { P ∈ P : x ∈ P } , x ∈ X ; st( A, P ) = ∪{ P ∈ P : A ∩ P 6 = ∅} , A ⊂ X ; P m = { P ∈ P : if P ⊂ Q ∈ P , then Q = P } . Readers ma y refer to [6, 13] for unstated definitions and terminology . 2. Regular Bases a t non -isola ted points Definition 2.1. Let B b e a base of a sp ace X . B is a r e gular b ase , see e.g. [6] ( r e g ular b ase at non-isolate d p oints , resp.) for X if for eac h (non-isolated, resp.) p oin t x ∈ X , B is regular at x . It is ob vious that regular bases ⇒ regular bases at non -isolated p oin ts, b u t regular bases at n on-isolated p oint s ; regular bases by Examp le 1.2. Definition 2.2. Let {W i } i ∈ N b e a sequ en ce of op en co v ers of a sp ace X and I ( X ) ⊂ S i ∈ N W i . { W i } i ∈ N is called a str ong development , see e.g. [6]( str ong development at non-isolate d p oints , resp.) for X if for every x ∈ X ( x ∈ X − I ) and eac h neigh b orho o d U of x there exist a neighborh o o d V of x and an i ∈ N su ch that st( V , W i ) ⊂ U . If {W i } i ∈ N is a strong d ev elopmen t at n on-isolated p oints, then so is {W i ∪ I ( X ) } i ∈ N . The follo wing Lemma 2.3 is pro v ed s im ilarly to Lemma 5.4.3 in [6], and lea ve to the reader the easy pro ofs of Lemm a 2.4 and 2.5. Lemma 2.3. If B is a r e gular b ase at non-isolate d p oints f or a sp ac e X , then the family B m ⊂ B is lo c al ly finite at non-isolate d p oint s and also c overs X − I . REGULAR BASES A T NON-ISOLA TED POINTS AND METRIZA TION THEOREMS 3 Lemma 2.4. L et B b e a r e gular b ase at non-isolate d p oints f or X . If B ′ ⊂ B is p oint- finite at non-isolate d p oints, then B ′′ = ( B − B ′ ) ∪ I ( X ) i s a r e gular b ase at non-isolate d p oints for X . Lemma 2.5. If B is a r e gular b ase at non-isolate d p oints for X , put B 1 = B m , B i = [( B − i − 1 [ j =1 B j ) ∪ I ( X )] m , i = 2 , 3 , · · · . Then B = ( S ∞ i =1 B i ) ∪ I ( X ) , and for e ach i ∈ N , B i is lo c al ly finite at non-isolate d p oints and B i +1 ∪ I ( X ) r efines B i ∪ I ( X ) . Recall that a top ological sp ace X is monotonic al ly normal [10] if for eac h ordered pair ( p, C ), where C is a closed set for X and p ∈ X − C , there exists an op en subset H ( p, C ) satisfying the follo wing conditions: (i) p ∈ H ( p, C ) ⊂ X − C ; (ii) F or ev ery closed subset D for X , if D ⊂ C , then H ( p, C ) ⊂ H ( p, D ); (iii) I f p 6 = q ∈ X , then H ( p, { q } ) ∩ H ( q , { p } ) = ∅ . A T 2 -paracompact sp ace or monotonically norm al space is a collecti on wise n ormal space [10]. Lemma 2.6. If a sp ac e X has a str ong development at non-isolate d p oints, then X is a monotonic al ly normal and p ar ac omp act sp ac e. Pr o of. Let {W i } i ∈ N b e a str ong d ev elopmen t at non-isolated p oint s for X , where W i +1 refines W i for ev ery i ∈ N . (1) Claim. Let A b e a closed subset for X . If x ∈ ( X − A ) ∩ ( X − I ), th en there exists an i ∈ N such that st( x, W i ) ∩ st( A, W i ) = ∅ . In fact, since X − A is an op en neighb orh o o d of x , th er e exists a j ∈ N and an op en neigh b orho o d V of x such that st ( V , W j ) ⊂ X − A . Also, there exists a i ≥ j such that st( x, W i ) ⊂ V . S in ce st( A, W i ) ⊂ X − V , we ha v e st( x, W i ) ∩ st( A, W i ) = ∅ . (2) X is a monotonically normal space. Let C b e a closed subset for X and p ∈ X − C . If p ∈ I , then w e let H ( p, C ) = { p } ; if p ∈ X − I , then there exists a min im um n ∈ N such that st( p, W n ) ∩ st( C , W n ) = ∅ b y (1), so we let H ( p, C ) = st( p, W n ). Th en H ( p, C ) is an op en subset for X . Clearly this definition of H ( p, C ) satisfies the conditions (i) and (ii) in the ab ov e defin ition of monotonically normal spaces. W e next prov e th at it also satisfies (iii). In fact, f or an y distinct p oints p, q in X − I , fix the n, m for whic h: H ( p, { q } ) = st( p, W n ) and H ( q , { p } ) = st( q , W m ) . Then st( p, W n ) ∩ st( q , W n ) = ∅ and st( p, W m ) ∩ st( q , W m ) = ∅ . By the c hoice of n, m , we hav e n = m , i.e, H ( p, { q } ) ∩ H ( q , { p } ) = ∅ . Hence it also satisfies (iii) in the d efinition of monotonicall y norm al sp aces. (3) X is a paracompact space. Let { G s } s ∈ S b e an op en co v er for X and S 0 = { s ∈ S : G s ∩ ( X − I ) 6 = ∅} . Fix a w ell-order b y “ < ” on S 0 . F or ev ery i ∈ N , s ∈ S 0 , put 4 FUCAI LIN, SHOU LIN, AND HEIKKI J. K. JUN NILA F s,i = X − (st( X − G s , W i ) ∪ ( S s ′ s ( x ), then G s ( x ) ∩ F s ′ ,i = ∅ , s o there is only one m em b er of { F s,i } s ∈ S 0 whic h meets G s ( x ) ∩ st( x, W i ). Hence { F s,i } s ∈ S 0 is a discrete and closed family for X . X is collect ion wise normal since monotonically n ormal spaces are collectio n wise n or- mal [10]. F or every F s,i , th ere exists an op en sub set G s,i suc h that F s,i ⊂ G s,i ⊂ G s and { G s,i } s ∈ S 0 is a discrete family . Let B i = { G s,i } s ∈ S 0 ∪ {{ x } : x ∈ I − [ s ∈ S 0 G s,i } . Then S i ∈ N B i is a σ -lo cally finite op en co v er for X and r efines { G s } s ∈ S . Since X is regular, X is paracompact. Next w e sh all pr o v e th e m ain th eorems in this s ection. Theorem 2.7. A sp ac e X has a r e g ular b ase at non-isolate d p oints if and only if X has a str ong development at non-isolate d p oints. Pr o of. Necessit y . Since X has a regular base at non-isolated p oints , X has a r egular base at non-isolated p oin ts B = ( S i ∈ N B i ) ∪ I ( X ) satisfying Lemma 2.5, where B i is lo cally fin ite at non-isolated p oints and B i +1 ∪ I ( X ) r efines B i ∪ I ( X ) for ev ery i ∈ N . Put W i = B i ∪ I ( X ). W e will sh o w that {W i } i ∈ N is a strong deve lopmen t at non -isolated p oints for X . In fact, for ev er y x ∈ X − I and eac h op en neigh b orho o d U of x , since B is r egular at non-isolated p oin ts , there exists an op en n eigh b orho o d V ⊂ U of x suc h that the set of all mem b ers of B that m eet b oth V and X − U is finite. W e can denote these finite element s by B 1 , B 2 , · · · , B k . Then there exists a j ∈ N such that B j ∩ { B i : i ≤ k } = ∅ . Hence st( V , W j ) ⊂ U . Sufficiency . L et {W i } i ∈ N b e a stron g develo pment at non-isolated p oin ts for X . By Lemma 2.6, X is paracompact. F or ev ery i ∈ N , let B i b e a lo cally fi nite op en refinement for W i . Without loss of generalit y , w e m a y assume B i +1 refines B i for ev ery i ∈ N . W e next p ro v e that B = ( S i ∈ N B i ) ∪ I ( X ) is a regular base at n on-isolated p oint s for X . Ob viously B is a b ase for X . F or ev ery x ∈ X − I and eac h op en neigh b orho o d U of x , there exist an op en neigh b orho o d V of x and an i ∈ N suc h th at st( V , W i ) ⊂ U . If j ≥ i , then st( V , B j ) ⊂ st( V , B i ) ⊂ st( V , W i ) ⊂ U. REGULAR BASES A T NON-ISOLA TED POINTS AND METRIZA TION THEOREMS 5 Ho w ever, since eac h B j is lo cally finite, there exists an op en neighborh o o d W ( x ) of x suc h that the set of all m emb ers of S j
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