## On locally compact Hausdorff spaces with finite metrizability number.(English)Zbl 1012.54002

The metrizability number $$m(X)$$ of a space X is the smallest cardinal number $$\kappa$$ such that $$X$$ can be represented as a union of $$\kappa$$ many metrizable subspaces. For example, the space $$\psi$$ (also called $${\mathcal N}\cup{\mathcal R}$$) has metrizability number 2, and its one-point compactification has metrizability number 3. The authors give more interesting examples. For one, they prove that the one-point compactification of any ladder system on $$\omega_1$$ has metrizability number 3. For another, they prove that for every $$3\leq n <\omega$$, there exists a compact separable Hausdorff space $$X$$ such that $$X$$ has weight $$\omega_1$$, metrizability number $$n$$ and $$X$$ is the increasing union of countably many closed subsets each of metrizability number 2. The main result is the theorem: If X is a locally compact Hausdorff space with $$m(X) =n<\omega$$, then for each $$1 \leq k<n$$, $$X$$ can be represented as $$X = G\cup F$$, where $$G$$ is an open dense subspace, $$m(G)=k$$, $$F=X\setminus G$$, and $$m(F)=n-k$$. It follows that by choosing $$k=n-1$$, the open dense set $$G$$ satisfies $$m(G) = n-1$$, and $$F= X\setminus G$$ is metrizable. The authors give some results on the metrizability number of the product of two spaces. For example, if $$X, Y$$ are Hausdorff spaces with $$X$$ compact and $$Y$$ locally compact, and both with finite metrizability number, then $$m(X\times Y)\geq (m(X) + m(Y) -1)$$. Some related topics are also considered. Among them are the metrizability number at a point, and the $$m$$-spectrum of a space. The paper continues a sequence of papers by the authors [Topology Appl. 59, No. 3, 287-298 (1994; Zbl 0840.54025); ibid. 63, No. 1, 69-77 (1995; Zbl 0860.54005) and ibid. 71, No. 2, 179-191 (1996; Zbl 0864.54001)].

### MSC:

 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54E35 Metric spaces, metrizability 54B10 Product spaces in general topology 54G20 Counterexamples in general topology

### Citations:

Zbl 0840.54025; Zbl 0860.54005; Zbl 0864.54001
Full Text:

### References:

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