By Artstein-Avidan S.

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**Additional info for A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators**

**Example text**

Put G = B(x, ), H = B(y, ) where = 12 d(x, y)) but fails in, for example, a trivial space (X, T0 ). What we do now is to see how ‘demanding certain minimum levels-of-supply of open sets’ gradually eliminates the more pathological topologies, leaving us with those which behave like metric spaces to a greater or lesser extent. 1 A topological space (X, T ) is T1 if, for each x in X, {x} is closed. 1 (i) T1 is hereditary 43 (ii) T1 is productive (iii) T1 ⇒ every finite set is closed. e. C is the weakest of all the T1 topologies that can be defined on X.

G. put G = B(x, ), H = B(y, ) where = 12 d(x, y)) but fails in, for example, a trivial space (X, T0 ). What we do now is to see how ‘demanding certain minimum levels-of-supply of open sets’ gradually eliminates the more pathological topologies, leaving us with those which behave like metric spaces to a greater or lesser extent. 1 A topological space (X, T ) is T1 if, for each x in X, {x} is closed. 1 (i) T1 is hereditary 43 (ii) T1 is productive (iii) T1 ⇒ every finite set is closed. e. C is the weakest of all the T1 topologies that can be defined on X.

Then there exists i0 ∈ I such that xi0 = yi0 in Xi0 . Choose disjoint open sets G, H in (Xi0 , Ti0 ) so that xi0 ∈ G, yi0 ∈ H. Then x ∈ πi−1 (G) ∈ T , y ∈ πi−1 (H) ∈ T and 0 0 −1 −1 since G ∩ H = ∅, πi0 (G) ∩ πi0 (H) = ∅. Hence result. The T2 axiom is particularly valuable when exploring compactness. Part of the reason is that T2 implies that points and compact sets can be ‘separated off’ by open sets and even implies that compact sets can be ‘separated off’ from other compact sets in the same way.