Countable Hausdorff spaces with countable weight. (English) Zbl 0588.54012

Let S be a commutative semigroup, C a class of objects in a category K with finite products. A mapping \(S\to C\), \(s\mapsto X(s)\), is called a productive represenation of S in C if (i) for any \(s_ 1,s_ 2\in S\) the objects \(X(s_ 1)\times X(s_ 2)\) and \(X(s_ 1+s_ 2)\) are isomorphic; (ii) \(X(s_ 1)\) is isomorphic with \(X(s_ 2)\) only if \(s_ 1=s_ 2\). In an earlier paper the author has shown that every countable commutative semigroup has a productive represenation in the class of all countable \(T_ 1\)-spaces with countable weight; the present paper improves this result by establishing the assertion for the class CHSCW of all countable \(T_ 2\) spaces with countable weight. (It is known that \(''T_ 2''\) cannot be replaced by \(''T_ 3''\), as is seen from the cyclic group of order two.) The proof also allows several additional conclusions; e.g. that \(({\mathbb{R}},+)\) has a productive representation in CHSCW.
Reviewer: W.Ruppert


54B10 Product spaces in general topology
20M30 Representation of semigroups; actions of semigroups on sets
20M14 Commutative semigroups
Full Text: EuDML