Pure Mathematics

The Structure of the Real Line by Lev Bukovský (auth.)

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By Lev Bukovský (auth.)

The speedy improvement of set concept within the final fifty years, often in acquiring lots of independence effects, strongly encouraged an realizing of the constitution of the true line. This publication is dedicated to the learn of the genuine line and its subsets taking into consideration the hot result of set concept. at any time when attainable the presentation is finished with no the total axiom of selection. because the booklet is meant to be self-contained, all useful result of set conception, topology, degree idea, descriptive set conception are revisited with the aim to do away with superfluous use of an axiom of selection. The duality of degree and classification is studied in a close demeanour. numerous statements touching on homes of the genuine line are proven to be undecidable in set conception. The metamathematics in the back of it really is almost immediately defined within the appendix. each one part features a sequence of routines with extra results.

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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.

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