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The Banach-Stone Theorem
... In metric spaces there is the notion of a (converging) sequence. However, this is not sufficient in general topological spaces. In this section we will discuss a well-known generalization. First of all, we have some definitions. 4.1. Definition. Let (X, T ) be a topological space and x ∈ X. A set N ...
... In metric spaces there is the notion of a (converging) sequence. However, this is not sufficient in general topological spaces. In this section we will discuss a well-known generalization. First of all, we have some definitions. 4.1. Definition. Let (X, T ) be a topological space and x ∈ X. A set N ...
The Stone-Cech compactification of Tychonoff spaces
... Theorem 5. Assume that X is a topological space and that fi : X → Xi , i ∈ I, are continuous functions. This family separates points from closed sets if and only if the collection of sets of the form fi−1 (Ui ), i ∈ I and Ui open in Xi , is a base for the topology of X. Proof. Assume that the family ...
... Theorem 5. Assume that X is a topological space and that fi : X → Xi , i ∈ I, are continuous functions. This family separates points from closed sets if and only if the collection of sets of the form fi−1 (Ui ), i ∈ I and Ui open in Xi , is a base for the topology of X. Proof. Assume that the family ...