On Mapping Sequences of a Projective Spectrum

Abstract

In the present work we will introduce two new notions, i.e., mapping sequence and projective point sequence, and by these notions, we study what one may call Alexandroff’s Homeomorphism Theorem¹. Namely by means of mapping sequences, we will divide that theorem into several homeomorphism theorems which clearly express the topological character of the mapping sequences, and by the use of projective point sequences, we simplify the proofs of these theorems. Then, we will give a proof of Borsuk’s² theorem by making use of the notion of mapping sequences. Finally, we consider an extension of the notion of a mapping sequence, and a few results are derived from this extension.

1 P. Alexandroff, Gestalt und Lage, Ann. of Math. 30 (1929), p. 134.

2 K. Borsuk, Sur les rétracts, Fund. Math. 17 (1931), p. 165.