By J. L. Dupont, I. H. Madsen

ISBN-10: 354009721X

ISBN-13: 9783540097211

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**Extra info for Algebraic Topology, Aarhus 1978**

**Sample text**

G. Pestov [24] proved: Assume that every finite product of spaces Xn , n = 1, 2, . . , is normal and countably paracompact. Then the σ -product of {Xn : n = 1, 2, . } is normal and countably paracompact. Kombarov [16] proved: Let σ be a σ -product of an uncountable number of spaces, each space having at least two points, and x ∈ σ . Then σ \ {x} is not normal. In particular, such a σ -product is not hereditarily normal. Later, Kombarov [17] improved this result to deduce that such a σ -product is not hereditarily pseudonormal and is not hereditarily countably paracompact.

He posed the question whether every normal space is countably paracompact or not. A normal space X is called a Dowker space if it is not countably paracompact, in other word, if X × I is not normal. E. Rudin constructed a Dowker space [KV, Chapter 17]. Thus Dowker’s problem was answered negatively. In 1976, concerning the normality of product spaces, K. Morita posed the following three conjectures [MN, Chapter 3]: M ORITA’ S CONJECTURE I. If X × Y is normal for any normal space Y , then X is a discrete space.

P. E. 4]. It is well known that every metric space is a paracompact p-space (due to Arhangel’ski˘ı) with countable tightness. , there exist a metric space T and a perfect map f : X → T . In 1978, Kombarov generalized Gul’ko–Rudin’s result by showing that every Σ-product of paracompact p-spaces {Xλ } is (collectionwise) normal if and only if all spaces Xλ have countable tightness [KV. 5]. A space X is said have the shrinking property if every open cover of X has a shrinking. If X is shrinking, then X is normal.

### Algebraic Topology, Aarhus 1978 by J. L. Dupont, I. H. Madsen

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