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On a question of Sierpiński

100%

Slaman T. A.

Fundamenta Mathematicae

1999

tom 159

nr 2

153-159

There is a set U of reals such that for every analytic set A there is a continuous function f which maps U bijectively to A.

Some examples of true $F_{σδ}$ sets

100%

Balcerzak M. , Darji U. B.

Colloquium Mathematicum

2000

tom 86

nr 2

203-207

Let K(X) be the hyperspace of a compact metric space endowed with the Hausdorff metric. We give a general theorem showing that certain subsets of K(X) are true $F_{σδ}$ sets.

Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance

80%

Jaeger P.

Formalized Mathematics

2014

tom 22

nr 3

199-204

We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets [16] (pp. 9-10). Literature [16] (pp. 9-10) and [11] (pp. 11-12) gives an overview, that there exists some other sets for this construction. Last we define special functions as random variables for stochastic finance in discrete time. The relevant functions are implemented in the article [15], see [9] (p. 4). The aim is to construct events and random variables, which can easily be used with a probability measure. See as an example theorems (10) and (14) in [20]. Then the formalization is more similar to the presentation used in the book [9]. As a background, further literatures is [3] (pp. 9-12), [13] (pp. 17-20), and [8] (pp.32-35).