 By Lev Bukovský (auth.)

The quick improvement of set concept within the final fifty years, regularly in acquiring lots of independence effects, strongly encouraged an knowing of the constitution of the genuine line. This e-book is dedicated to the research of the true line and its subsets bearing in mind the hot result of set idea. at any time when attainable the presentation is completed with out the entire axiom of selection. because the e-book is meant to be self-contained, all worthy result of set idea, topology, degree thought, descriptive set concept are revisited with the aim to cast off superfluous use of an axiom of selection. The duality of degree and classification is studied in an in depth demeanour. a number of statements referring to houses of the genuine line are proven to be undecidable in set thought. The metamathematics in the back of it's presently defined within the appendix. every one part includes a sequence of routines with extra results.

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Z is an integral domain and Q is the ﬁeld of fractions of Z. The set Z being a union of two countable sets is a countable set. The set Q is the image of the countable set N × Z by the mapping f (n, z) = z/(n + 1), therefore Q is countable. 6) Consequently the set R \ Q of irrationals has cardinality c. 7. For any reals a < b there exists a rational r such that a < r < b. Proof. We can assume that 0 ≤ a < b (the other cases can be easily reduced to that one). 1. The Deﬁnition 43 1/n < b − a. Using again Archimedes’ Principle there exists a natural number m such that m · 1/n > a.

23) holds true. However the condition that S is the class of all topological spaces is essential. We show it. Let S denote the class of all topological spaces, ϕ being the property (A ⊆ X is inﬁnite) ∧ ( X, O is compact). 23) fails. 23) holds true. In such a case we emphasize the fact that an introduced property of a set is internal for a class S in a theory T. The topologists usually distinguish between internal and external property by speaking about a property of a topological space and a property of a set, respectively8 .

A(Vn )} a ﬁnite subcover of U. d) Prove in ZF that a topological product of ﬁnitely many compact spaces is a compact space. 19 Countability Axioms If for any x ∈ X, the neighborhood ﬁlter N (x) has a countable base, then we say that X, O satisﬁes the ﬁrst axiom of countability or is ﬁrst countable. If there exists a countable base of the topology O, then we say that the space X, O satisﬁes the second axiom of countability or is second countable. X is Lindel¨ of if any open cover of X contains a countable subcover.