Geodesic
On Unions of Metrizable Subspaces
Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 76-85

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we study the question “which generalized metric spaces can be written as the union of k (closed) metrizable subspaces, where k is a cardinal number with k ≤ c?“ Questions of this type first arose in [16] where J. Nagata asked for examples of certain generalized metric spaces which could not be written as the union of countably many closed metrizable subspaces. Using Baire Category arguments, Fitzpatrick provided the required examples in [12]. We begin this paper by sharpening Fitzpatrick's examples, showing in Section 2 that there is a Moore space which is not the union of countably many metrizable subspaces of any kind. Then in Section 3 we present a positive result, proving that any cr-space, and a fortiori any Moore space, can be written as the union of c = 2ω0 closed metrizable subspaces. 
Douwen, E. K. Van; Lutzer, D. J.; Pelant, J.; Reed, G. M. On Unions of Metrizable Subspaces. Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 76-85. doi: 10.4153/CJM-1980-008-5
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