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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

##### MSC:
 54B10 Product spaces in general topology 20M30 Representation of semigroups; actions of semigroups on sets 20M14 Commutative semigroups
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