Uniform covers at non-isolated points

📝 Abstract

In this paper,\ the authors define a space with an uniform base at non-isolated points, give some characterizations of images of metric spaces by boundary-compact maps, and study certain relationship among spaces with special base properties.\ The main results 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-isolated points; (2)\ Each discretizable space of a space with an uniform base is an open compact and at most boundary-one image of a space with an uniform base; (3)\ $X$ has a point-countable base if and only if $X$ is a bi-quotient,\ at most boundary-one and countable-to-one image of a metric space.

💡 Analysis

In this paper,\ the authors define a space with an uniform base at non-isolated points, give some characterizations of images of metric spaces by boundary-compact maps, and study certain relationship among spaces with special base properties.\ The main results 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-isolated points; (2)\ Each discretizable space of a space with an uniform base is an open compact and at most boundary-one image of a space with an uniform base; (3)\ $X$ has a point-countable base if and only if $X$ is a bi-quotient,\ at most boundary-one and countable-to-one image of a metric space.

📄 Content

arXiv:1106.4133v1 [math.GN] 21 Jun 2011 UNIFORM COVERS AT NON-ISOLATED POINTS FUCAI LIN AND SHOU LIN Abstract. In this paper, the authors define a space with an uniform base at non-isolated points, give some characterizations of images of metric spaces by boundary-compact maps, and study certain relationship among spaces with special base properties. The main results 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-isolated points; (2) Each discretizable space of a space with an uniform base is an open compact and at most boundary-one image of a space with an uniform base; (3) X has a point-countable base if and only if X is a bi-quotient, at most boundary-one and countable-to-one image of a metric space.

1. Introduction Topologists obtained many interesting characterizations of the images of metric spaces by some kind of maps. A. V. Arhangel’skiˇı [3] proved that a space X is an open compact image of a metric space if and only if X has an uniform base. Recently, C. Liu [16] gives a new characterization of spaces with a point-countable base by pseudo-open and at most boundary-one images of metric spaces. How to character an open or pseudo-open and boundary-compact images of metric spaces? On the other hand, a study of spaces with a sharp base or a weakly uniform base [5, 6] shows that some properties of a non-isolated point set of a topological space will help us discuss a whole construction of a space. In this paper, the authors analyze some base properties on non-isolated points of a space, introduce a space having an uniform base at non-isolated points and describe it as an image of a metric space by open boundary-compact maps. Some relationship among the images of metric spaces under open boundary-compact maps, pseudo-open boundary-compact maps, open compact maps, and spaces with a point-countable base are discussed. By R, N, denote the set of real numbers and positive integers, respectively. For a space X, let I(X) = {x : x is an isolated point of X} and I(X) = {{x} : x ∈I(X)}. In this paper all spaces are T2, all maps are continuous and onto. Recalled some basic definitions. Let X be a topological space. X is called a metacompact (resp. paracompact, meta-Lindel¨of) space if every open cover of X has a point-finite (resp. locally finite, point-countable) open refinement. X is said to have a Gδ-diagonal if the diagonal 2000 Mathematics Subject Classification. 54C10; 54D70; 54E30; 54E40. Key words and phrases. Boundary-compact maps; developable spaces; uniform bases; sharp bases; open maps; pseudo-open maps. Supported in part by the NSFC(No. 10571151). 1 2 FUCAI LIN AND SHOU LIN ∆= {(x, x) : x ∈X} is a Gδ-set in X × X. X is called a perfect space if every open subset of X is an Fσ-set in X. Definition 1.1. Let P be a base of a space X. (1) P is an uniform base [1] (resp. uniform base at non-isolated points) for X if for each (resp. non-isolated) point x ∈X and P′ is a countably infinite subset of (P)x, P′ is a neighborhood base at x. (2) P is a point-regular base [1] (resp. point-regular base at non-isolated points) for X if for each (resp. non-isolated) point x ∈X and x ∈U with U open in X, {P ∈(P)x : P ̸⊂U} is finite. In the definition, “at non-isolated points” means “at each non-isolated point of X”. It is obvious that uniform bases (resp. point-regular bases)⇒uniform bases at non-isolated points (resp. point-regular bases at non-isolated points), but we will see that uniform bases at non-isolated points (resp. point-regular bases at non-isolated points) ̸⇒uniform bases (resp. point-regular bases) by Example 4.1. Definition 1.2. Let X be a space, and {Pn} a sequence of open subsets of X. (1) {Pn} is called a quasi-development [8] for X if for every x ∈U with U open in X, there exists n ∈N such that x ∈st(x, Pn) ⊂U. (2) {Pn} is called a development (resp. development at non-isolated points) for X if {st(x, Pn)}n∈N is a neighborhood base at x in X for each (resp. non-isolated) point x ∈X. (3) X is called quasi-developable (resp. developable, developable at non-isolated points) if X has a quasi-development (resp. development, development at non-isolated points). It is obvious that every development for a space is a development at non-isolated points, but a space having a development at non-isolated points may not have a development, see Example 4.2. Definition 1.3. Let f : X →Y be a map. (1) f is a compact map (resp. s-map) if each f −1(y) is compact (resp. separa- ble) in X; (2) f is a boundary-compact map (resp. boundary-finite map, at most boundary- one map) if each ∂f −1(y) is compact (resp. finite, at most one point) in X; (3) f is an open map if whenever U open in X, then f(U) is open in Y ; (4) f is a bi-quotient map (resp. countably bi-quotient map) if for any y ∈Y and any (resp. countable) family U of open subsets in X with f −1(y) ⊂∪U, there exists finite subset U′ ⊂U such that y ∈Intf(∪U′); (5) f is a pseudo-open map if whenever f −1(

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