Uniform bases at non-isolated points and maps
In this paper, the authors mainly discuss the images of spaces with an uniform base at non-isolated points, and obtain the following main results: (1)\ Perfect maps preserve spaces with an uniform base at non-isolated points; (2)\ Open and closed map…
Authors: Fucai Lin, Shou Lin
UNIF ORM BASES A T NON-ISOLA TED POINTS AND MAPS FUCAI LIN AND SHOU LIN Abstract. I n this paper, the authors mainly discuss the images of s paces with an uniform base at non-isolated points, and obtain the following main results: (1) Pe rfect maps preserv e spaces wi th an uniform base at non-isolated points; (2) Open and closed maps pr eserv e r egular spaces wi th an unifor m base at non-isolated p oints; (3) Spaces with an uniform base at non-isolated points don’t satisfy the decomposition theorem. 1. Intr oduction Recently , s pa ces with an uniform base or spaces with a s harp base bring some top ologist a ttention and in teresting results ab out certain bases a re o btained [2, 3, 14]. In [9], the a uthors define the no tion o f unifor m bases at no n- isolated p o ints and o btain some related matters. F o r example, it is proved that a space X ha s a n uniform base at non-isolated po ints if and only if X is the op en b oundary -compact image o f a metric space . It is well k nown that the cla ss of spa ces under the op en and compact images of metric spa ces are preserved b y perfect maps o r closed and op en maps(se e [14]). Hence a question arise s :“What k ind o f maps pr e serve spaces with a uniform base at non-isola ted points?” In this pap er we shall consider the inv ariance of spaces with an unifor m ba se at non-isolated p oints under per fect ma ps or clo s ed a nd o p en maps. By R , N , denote the set of all r eal num b ers and p ositive int eger s, resp ectively . F o r a top o logical space X , let τ ( X ) denote the top o lo gy for X , and let I ( X ) = { x : x is an iso lated p o int of X } , X d = X − I ( X ) , I ( X ) = { { x } : x ∈ I ( X ) } , I ∆ ( X ) = { ( { x } , { x } ) : x ∈ I ( X ) } . In this pap er all spac e s are Ha usdorff, all maps a re contin uous and onto. Recall some basic definitions. Definition 1. 1 . Let P b e a base of a space X . P is an un iform b ase [1] (resp. uniform b ase at non- isolate d p oints [9]) for X if for each ( resp . non-isolated) p oint x ∈ X and P ′ is a countably infinite subs e t o f { P ∈ P : x ∈ P } , P ′ is a neighborho o d base at x in X . In the definition, “a t non-isola ted po int s” mea ns “ a t each non-isolated p oint of X ”. 2000 Mathematics Subje ct Classific ation. 54 C10; 54D70; 54E30; 54E40. Key wor ds and phr ases. P erfect mappings; uniform bases at non-isolated p oints; op en map- pings; dev elopable at non-isolated p oints. Supported in part by the NSFC( No. 10571151). 1 2 FUCAI LIN AND S HO U LIN Definition 1.2. [8] Let f : X → Y b e a map. (1) f is a b oundary-c omp act map , if ea ch ∂ f − 1 ( y ) is compact in X ; (2) f is a c omp act map if each f − 1 ( y ) is compact in X ; (3) f is a p erfe ct map if f is a clos ed a nd compact map. Definition 1.3. Let X b e a space and {P n } n a sequence of collections of op en subsets o f X . (1) {P n } n is called a quasi-development [4] for X if for e very x ∈ U with U op en in X , ther e ex is ts n ∈ N such that x ∈ s t( x, P n ) ⊂ U . (2) {P n } n is called a development [13](resp. development at non-isolate d p oints [9]) for X if { st( x, P n ) } n is a neighborho o d base at x in X for each (r esp. non- isolated) po int x ∈ X . (3) X is ca lled quasi-develop able (resp. develo p able , develop able at non-isolate d p oints ) if X has a quasi-developmen t (resp. dev elopment, dev elopment at non-isolated p oints). Obviously , in the definition a b out developments at no n-isolated points we can assume that each P n is a cov er for X . Also, it is easy to s e e that a spa ce which is developable at non-is olated points is quasi-developable, but a space with a devel- opment a t non-isola ted p oints may not hav e a developmen t, see Example in [9]. Definition 1.4. Let P b e a family of subsets of a spa ce X . P is called p oint-finite at non-isolate d p oints [9] if for each non-isolated po int x ∈ X , x belongs to a t most finite elements of P . Let {P n } n be a developmen t (r esp. a developmen t at non-isolated po int s) for X . {P n } n is said to b e a p oint-finite development (res p. a p oint-finite development at non- isolate d p oints ) for X if each P n is po int -finite a t each (re s p. non-isolated) p o int o f X . Readers may refer to [8, 10] for unstated definitions and terminolog y . 2. Developments a t non-isol a ted points In this se c tion some characteriz a tions of spaces with a developmen t at no n- isolated p oints are established. Let X b e a top o logical space. g : N × X → τ ( X ) is called a g - function, if x ∈ g ( n, x ) and g ( n + 1 , x ) ⊂ g ( n, x ) for a ny x ∈ X and n ∈ N . F or A ⊂ X , put g ( n, A ) = [ x ∈ A g ( n, x ) . Theorem 2.1. L et X b e a top olo gic al sp ac e. Then the f ol lowing c onditions ar e e quivalent: (1) X has a development at n on-isolate d p oints; (2) The r e exists a g -function for X such that, for every x ∈ X d and se quenc es { x n } n , { y n } n of X , if { x, x n } ⊂ g ( n, y n ) for every n ∈ N , then x n → x. (3) X is a qu asi-develop able sp ac e, and X d is a p erfe ct subsp ac e of X . Pr o of. (1) ⇒ (2) . Let {U n } n be a developmen t a t non-isola ted p oints for X . W e can a ssume that I ( X ) ⊂ U n for every n ∈ N . F o r every x ∈ X a nd n ∈ N , fix U n ∈ U n with x ∈ U n , wher e U n = { x } when x ∈ I ( X ). Le t g ( n, x ) = T i
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