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N. It’s easy to check that yp → (l1 , l2 , . . , ln ) so that X is sequentially compact, as required. ’ Proof Let (X, T ) = i∈I (Xi , Ti ); let ∅ ⊂ Yi ⊆ Xi for each i ∈ I. There appear to be two different ways to topologise Yi : either (i) give it the subspace topology induced by Ti or (ii) give it the product of all the individual subspace topologies (Ti )Yi . e. G∗i0 = Yi0 ∩ Gi0 for some Gi0 ∈ Ti0 , a typical subbasic open set for (ii) is {(yi ) ∈ Yi : yi0 ∈ G∗i0 } which equals Yi ∩ {(xi ) ∈ = Xi : xi0 ∈ Gi0 ∈ Ti0 , i0 ∈ I} Yi ∩ { a typical open cylinder in Xi } which is a typical subbasic open set in (i).

4 Every T4 space is T3 1 . 1. 8 Any compact T2 space is T4 . 3. Note Unlike the previous axioms, T4 is neither hereditary nor productive. The global view of the hierarchy can now be filled in as an exercise from data supplied above:Metrizable Hereditary? Productive? T4 T3 1 2 T3 T2 T1 The following is presented as an indication of how close we are to having ‘come full circle’. 9 Any completely separable T4 space is metrizable! Sketch Proof Choose a countable base; list as {(Gn , Hn ) : n ≥ 1} those pairs of elements 49 of the base for which G¯n ⊆ Hn .

Thus there exists a subnet (zγ ) of (yβ ) in F which converges in X, whence its limit is in F ). 2 Compactness is preserved by continuous maps Proof (for if X is compact and f continuous, let (yα )α∈A be a net in f (X). Then for each α ∈ A, yα = f (xα ) for some xα ∈ X. The net (xα )α∈A has a convergent subnet (zβ )β∈B , say zβ → l, whence f (zβ ) → f (l). Then (f (zβ ))β∈B is a convergent subnet of (yα )α∈A ). Example If (xnk ) is a subsequence of a sequence (xn ), then it is a subnet of it also; because the ith 0 tail of the sequence (xn ) is {xi0 , xi0 +1 , xi0 +2 , .

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