Some remarks concerning C-spaces. (English) Zbl 1116.54018

Borst, Piet

Topology Appl. 154, No. 3, 665-674 (2007).

Basically this paper is concerned with various types of C-spaces together with results involving such well known concepts from infinite dimension theory such as small weakly infinite-dimensional spaces (small w.i.d.), weakly infinite-dimensional spaces in the sense of Alexandrov (Smirnov), (A-w.i.d. (S-w.i.d.)) and countable dimensional spaces (c.d.) along with some notions of disconnectedness. A space \(X\) is called a (finite) C-Ha space (in the sense of Haver) iff there exists an admissible metric \(d\) on \(X\) such that for every sequence \(\{\varepsilon_i\}\), \(\varepsilon_i> 0\), there exist disjoint open families \(V_i\) \((i= 1,\dots, n)\), \(i= 1,2,\dots\), such that \(X\) is the union of the \(V_i\), and \(\sup\{d(x, y): x\), \(y\) in \(v\), \(v\) in \(V_i\}< \varepsilon_i\), for each \(i\). A space \(X\) is called a (finite) C-space iff for every sequence of (finite) open covers \(U_i\), there exist disjoint open families \(V_i\) \((i= 1,\dots, n)\), \(i= 1,2,\dots\) such that \(X\) is the union of the \(V_i\) and for all \(i\), each element of \(V_i\) is contained in an element of \(U_i\).
Among the many and varied results of the paper the following are representative:
(1) Each C-space is a C-Ha space and these notions coincide for compact spaces.
(2) Every c.d. space is a C-space and every C-space is A-w.i.d.
(3) \(X\) is a finite C-Ha space iff \(X\) admits a metric compactification which is a C-space iff for some admissible metric \(d\) on \(X\) the metric dimension of \((X, d)\) exists.
(4) Every space having small transfinite dimension and every complete strongly \(\sigma\)-totally disconnected space is a finite C-Ha space, and every finite C-Ha space is small w.i.d.
(5) Every space having large transfinite dimension is a finite C-space and every finite C-space is S-w.i.d.

Reviewer: Marvin V. Mielke (Coral Gables)

MSC:

54F45

Dimension theory in general topology

Keywords:

compactification; C-space; finite C-space; totally disconnected

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