On projective finite spaces

A.V. Arhangel’skii [1] defines the notion of projective power of a Tychonoff space. In particular, the space X is projectively finite, if for any open continuous mapping : → onto metrizable separable space the image = 𝑓(𝑋) is a finite set. The following results are obtained in this paper: Теорема 1. There is a projectively finite metrizable space of the weight = 2ℵ0. Теорема 2. For every cardinal number ≥ 2ℵ0 there exists such a projectively finite Tychonoff space 𝑋, that any Tychonoff space of the weight ≤ is embeddable in 𝑋. It follows from theorems 1 and 2 the existance of such a projectively finite space, that contains a non trivial convergent sequence. This is an affirmative answer to one of the question of A.V. Arhangel’skii [1].