Connected extensions of metric spaces and fixed points of Banach contractions

Two theorems announced by the author are proved. The first theorem concerns the existence of connected metrizable extensions of metric spaces. The second theorem is a slightly paradoxical assertion about the validity of the fixed-point principle. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 227–237.