By Lev Bukovský (auth.)
The fast improvement of set idea within the final fifty years, as a rule in acquiring lots of independence effects, strongly prompted an realizing of the constitution of the genuine line. This e-book is dedicated to the research of the true line and its subsets making an allowance for the new result of set thought. every time attainable the presentation is finished with no the entire axiom of selection. because the e-book is meant to be self-contained, all beneficial result of set concept, topology, degree thought, descriptive set idea are revisited with the aim to put off superfluous use of an axiom of selection. The duality of degree and classification is studied in a close demeanour. numerous statements touching on houses of the genuine line are proven to be undecidable in set concept. The metamathematics in the back of it's almost immediately defined within the appendix. each one part features a sequence of routines with extra results.
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Hint: λ (κ+ ) = ξ<κ+ λ ξ. g) Prove the Bukovsk´ y-Hechler formula: if cf(κ) < κ, λ < κ and 2µ = 2λ for every κ λ ≤ µ < κ, then 2 = 2λ . κξ Hint: If κ = ξ Xk ) holds true for any x ∈ N. 1. Using the First Theorem on Mathematical Induction one can prove basic properties of natural numbers. 2. , 0 ≤ n for any n ∈ N. b) If n is a natural number, n = 0, then there exists a natural number m such that n = m + 1. c) If m < n, then m + 1 ≤ n. d) If n, m are natural numbers such that |n − m| < 1, then n = m. e) If m ≤ n, then there exists a natural number k such that n = m + k. f) If n, m are natural numbers, then also n + m, n · m are natural numbers. Proof. If f : X1 −→ X2 , then f ⊆ X1 × X2 can be considered as the graph of f . 38. Assume that X1 , O1 , X2 , O2 are Hausdorff topological spaces and f : X1 −→ X2 is continuous. Then f is a closed subset of X1 × X2 . A convergence structure on a set X is a mapping lim : X −→ X from a set X ⊆ ω X. A sequence {xn }∞ n=0 belonging to X is called convergent and the value lim({xn }∞ n=0 ) is called the limit of it and denoted limn→∞ xn . A set X endowed 28 Chapter 1. Introduction with a convergence structure lim is called an L∗ -space if the following conditions are satisfied: (L1) (L2) (L3) if xn = x for every n, then limn→∞ xn = x; if limn→∞ xn = x and {nk }∞ k=0 is increasing, then limk→∞ xnk = x; if x = limn→∞ xn , then there exists a subsequence {xnk }∞ k=0 such that no subsequence of {xnk }∞ k=0 has limit x.