A second course in general topology by Heikki Junnila

By Heikki Junnila

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Hence the collection {kα : α ∈ A} is locally finitely supported, and it follows that the function k = α∈A kα is continuous. If we set gα = kα /k for each α ∈ A, then we get a partition of unity {gα : α ∈ A} with the required properties. 4. Continuous selections. In this section, we shall use partitions of unity to prove a fundamental result in the “theory of selections”. We need some definitions before can state the result. Let ϕ be a mapping from a space X into the family of all non-empty subsets of a space Y .

As a consequence, we have that, for every x ∈ X, the sequence St(x, Vn ) n∈N is a nbhd base at x. Let n ∈ N. We show that St2 (x, Vn+1 ) ⊂ St(x, Vn ) for each x ∈ X. Let x ∈ X and z ∈ St2 (x, Vn+1 ). Then there exists O, U ∈ Vn+1 such that x ∈ O, z ∈ U and O ∩ U = ∅. Let i, j ∈ N and V, W ∈ Vn be such that we have O ∈ Ui , U ∈ Uj , St(O, Ui ) ⊂ V and St(U, Uj ) ⊂ W . Now if i ≤ j, then O ∪ U ⊂ St(O, Ui ) ⊂ V , and if j ≤ i, then O ∪ U ⊂ St(U, Uj ) ⊂ W . As a consequence, we have either that {x, z} ⊂ V or that {x, z} ⊂ W , and in both case we have that z ∈ St(x, Vn ).

Let X be a T1 -space with a normal closed base N . It follows from Propositions 9 - 11 that the space UN is compact and Hausdorff, and X can be embedded in UN . As a consequence, the space X is Tihonov. Since every Tihonov space has a normal closed base, we have proved the following purely topological characterization of the Tihonov property. 14 Theorem A T1 -space is a Tihonov space iff the space has a normal closed base. Let X be a Tihonov space. Let Z = ZX , and denote the subspace {Zx : x ∈ X} of ˜ By Propositions 11 and 13, the space X ˜ is homeomorphic with X and UZ is a UZ by X.

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