Mathematica Moravica, Vol. 16-1 (2012)

NOTE: Vol. 16-1 (2012) of Mathematica Moravica is dedicated to Prof. MILAN R. TASKOVIĆ on the occasion of his retirement.

The Axiom of Infinite Choice
Mathematica Moravica, Vol. 16-1 (2012), 1–32.

Abstract. In this paper we present the Axiom of Infinite Choice: Given any set $P$, there exist at least countable choice functions or there exist at least finite choice functions.
This paper continues the study of the Axiom of Choice by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Tasković [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Fredholm and Leray-Schauder alternatives are two direct consequences of the Axiom of Infinite Choice.
Keywords. The Axiom of Infinite Choice, The Axiom of Choice, Zermelo's Axiom of Choice, Lemma of Infinite Maximality, Zorn's lemma, Restatements of the Axiom of Infinite Choice, Choice functions, Foundation of the Fixed Point Theory, Geometry of the Axiom of Infinite Choice, Axioms of Infinite Choice for Points and Apices.

On a result of Jachymski, Matkowski, and Świątkowski
Mathematica Moravica, Vol. 16-1 (2012), 33–35.

Keywords. Fixed point, Nonlinear conditions for fixed points, topological spaces, Monotone Principle of Fixed Point, TCS-convergence, $d$-Cauchy completeness.