## Fractals Everywhere, Second Edition by Michael F. Barnsley

By Michael F. Barnsley

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Proof. Set F (x) = {U ∈ B : x ∈ U } ∈ P(B). If X is Hausdorﬀ, then F is an injection. A selector of a countable base B of X is a countable dense subset of X. , if x ∈ V ⊆ U for some open set V . One can easily show that x ∈ A ≡ A ∩ U = ∅ for any neighborhood U of x. 17) Let V be a family of non-empty subsets of X, x ∈ X. V is called a neighborhood base at the point x if a set V is a neighborhood of x if and only if there exists a set U ∈ V such that U ⊆ V . A point x ∈ X is called an accumulation point of A if for every neighborhood U of x the intersection U ∩ A has at least two points.

So, let x∈ / A. We want to show that x ∈ / A. Set W = {U ∈ O : x ∈ r(U )}. If y ∈ A, then x = y and therefore there are open sets U, V such that y ∈ U , x ∈ V and U ∩ V = ∅. Then V ⊆ r(U ) and therefore U ∈ W. Thus W is an open cover of A. Since A is a compact set, there exist ﬁnitely many U0 , . . , Un ∈ W such that A ⊆ U0 ∪ · · · ∪ Un . Then r(U0 ) ∩ · · · ∩ r(Un ) is a neighborhood of x disjoint with the set A, consequently x ∈ / A. 2. Topological Preliminaries 25 Now assume that X is a compact space and A is a closed subset of X, x ∈ / A.

The implication c) → a) is trivial. So the equivalences a) ≡ b) ≡ c) can be proved in ZF. For a proof of c) → d) one needs wAC. The condition a), therefore any of its equivalent conditions b), c) and d), is called the sequence selection property, shortly SSP. 41. The convergence structure of a Fr´echet topological space of cardinality c possesses the sequence selection property. If X is a topological group, then X possesses SSP if and only if the following condition e) is satisﬁed: e) If limm→∞ xn,m = e for every n, then there exist increasing sequences {nk }∞ k=0 and {mk }∞ k=0 such that limk→∞ xnk ,mk = e.