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

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.

### MSC:

 54F45 Dimension theory in general topology

### Keywords:

compactification; C-space; finite C-space; totally disconnected
Full Text:

### References:

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