## Spaces having a compactification which is a $$C$$-space.(English)Zbl 1055.54009

The authors characterize those separable metrizable spaces that can be densely embedded into some compact metrizable $$C$$-space. A space $$X$$ is called a $$C$$-space provided for every sequence $$(\mathcal{C}_n)$$ of open covers of $$X$$ there exists a sequence $$(\mathcal{H}_n)$$ of collections of pairwise disjoint open subsets of $$X$$ such that $$\bigcup_n\mathcal{H}_n$$ covers $$X$$ and $$\mathcal{H}_n$$ refines $$\mathcal{C}_n$$ for each $$n\in\mathbb{N}$$. The characterization is in terms of the existence of suitable bases for $$X$$.

### MSC:

 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54F45 Dimension theory in general topology

### Keywords:

compactification; C-space; base
Full Text:

### References:

 [1] Addis, D.F; Gresham, J.H, A class of infinite-dimensional spaces, part I: dimension theory and Alexandroff’s problem, Fund. math., 101, 195-205, (1978) · Zbl 0397.54051 [2] Borst, P, Spaces having a weakly-infinite-dimensional compactification, Topology appl., 21, 261-268, (1985) · Zbl 0587.54055 [3] P. Borst, Some remarks concerning C-spaces, Preprint · Zbl 1116.54018 [4] Chatyrko, V.A, Classification of metric compacta with property C, Questions answers gen. topology, 10, 51-62, (1992) · Zbl 0752.54011 [5] Chatyrko, V.A, Classification of compacta with property C, Siberian math. J., 33, 1144-1148, (1992) · Zbl 0834.54018 [6] Chatyrko, V.A, On factorization theorem for transfinite dimension dim_{C}, Sci. math., 3, 357-366, (2000) · Zbl 1004.54022 [7] Engelking, R, Theory of dimensions, finite and infinite, (1995), Heldermann Berlin · Zbl 0872.54002 [8] Engelking, R; Pol, E, Countable-dimensional spaces: A survey, Dissertationes math., 216, (1983) · Zbl 0496.54032 [9] Engelking, R; Pol, R, Compactifications of countable-dimensional and strongly countable-dimensional spaces, Proc. amer. math. soc., 104, 985-987, (1988) · Zbl 0691.54015 [10] Haver, W.E, A covering property for metric spaces, (), 108-113 [11] Misra, A.K, Some regular wallman βX, Indag. math., 35, 237-242, (1973) · Zbl 0258.54022 [12] Pol, R, A weakly infinite-dimensional compactum which is not countable-dimensional, Proc. amer. math. soc., 82, 634-636, (1981) · Zbl 0469.54014 [13] Schurle, A.W, Compactification of strongly countable-dimensional spaces, Trans. amer. math. soc., 136, 25-32, (1969) · Zbl 0175.19902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.