By Satya Deo

ISBN-10: 8185931682

ISBN-13: 9788185931685

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Additional info for Algebraic Topology: A Primer (Texts and Readings in Mathematics)

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In [110], Cech introduced the following definition. A topological space X is called perfectly normal if X is normal and every open subset of X is an Fσ -set. 2. It turns out that every perfectly normal space X is completely normal, that is, every subset Y ⊂ X is normal (see [18 exerc. 7, 9 and 11 p. IX. 102–103], [64, 111]). 3 can be extended to perfectly normal spaces. Consequently, every subset Y of a perfectly normal space X satisfies dim(Y ) ≤ dim(X ) [111, par. 28]. Alexandroff (see [8, p.

This shows in particular that the set R\Q of irrational numbers satisfies dim(R\Q)=0. 22, we obtain the following results. 24 Let (X i )i∈I be a family of compact Hausdorff spaces with dim(X i ) = 0 for all i ∈ I . Then the product space X := i∈I X i satisfies dim(X ) = 0. 7, the space X is scattered since it is a product of scattered spaces. On the other hand, X is a product of compact Hausdorff spaces and hence also compact and Hausdorff. 25 Let (X i )i∈I be a family of non-empty finite discrete spaces.

1 We say that a topological space X is totally separated if the quasicomponent of every point x ∈ X is the singleton set reduced to the point x. 2 A topological space X is totally separated if and only if it satisfies the following condition: for every pair of distinct points x and y in X , there exists a partition of X into two open subsets U and V such that x ∈ U and y ∈ V . 3 Every totally separated space is Hausdorff. Proof This immediately follows from the preceding remark. 4 Let X be a topological space and x a point in X .

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Algebraic Topology: A Primer (Texts and Readings in Mathematics) by Satya Deo


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