The Alexandroff Duplicate and its subspaces - RiuNet
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The Alexandroff Duplicate and its subspaces - RiuNet
Agata Casertaâ and Stephen Watson. Abstract. We study some ... potential for finding. Michael spaces among the subspaces of Alexandroff duplicates. ... Define a base for a topology on Y = X à 2 by B = B0 ⪠B1, where. B0 is the family of all ...
Applied General Topology
@
c Universidad Polit´
ecnica de Valencia Volume 8, No. 2, 2007 pp. 187-205
The Alexandroff Duplicate and its subspaces Agata Caserta∗ and Stephen Watson
Abstract. We study some topological properties of the class of the Alexandroff duplicates and their subspaces. We give a characterization of metrizability and Lindel¨ of properties of subspaces of the Alexandroff duplicate. This characterization clarifies the potential for finding Michael spaces among the subspaces of Alexandroff duplicates.
2000 AMS Classification: 54B05, 54B10. Keywords: type line.
Resolution, Alexandroff duplicate, Lindel¨ of property, Michael-
In their famous 1922 memoir on compact spaces [1], Alexandroff and Urysohn defined a topological space that has become known as the Alexandroff double circle or the Alexandroff duplicate. In this paper we study several version of the Alexandroff duplicate by viewing it as particular resolution by constant maps. Alexandroff duplicates have been studied and used by many topologists. In particular, Michael’s 1963 example of a Michael spaces is subspace of an Alexandroff duplicate. This paper is an organized study of the topological properties of the class of the Alexandroff duplicates and of their subspaces. In particular, we characterize when subspaces of Alexandroff duplicates have the Lindel¨ of property. This suggests that the potential for finding Michael spaces among the subspaces of Alexandroff duplicates is not high. In this note, P and Q denote the set of the irrational and rational numbers, respectively. Ordinal numbers are denoted by Greek letters; when viewed as topological spaces, they are given the order topology. Products of topological spaces are endowed with the standard product topology. The symbol [A]λ denotes the family of subsets of A having size exactly λ. The symbols [A]≤λ and [A]
