Every metric space of weight $\lambda=\lambda^{\aleph_0}$ admits a condensation onto a Banach space

Abstract

In this paper, we have proved that for each cardinal number $\lambda$ such that $\lambda=\lambda^{\aleph_0}$ a metric space of weight $\lambda$ admits a bijective continuous mapping onto a Banach space of weight $\lambda$. Then, we get that every metric space of weight continuum admits a bijective continuous mapping onto the Hilbert cube. This resolves the famous Banach's Problem (when does a metric (possibly Banach) space $X$ admit a bijective continuous mapping onto a compact metric space?) in the class of metric spaces of weight continuum. Also we get that every metric space of weight $\lambda=\lambda^{\aleph_0}$ admits a bijective continuous mapping onto a Hausdorff compact space. This resolves the Alexandroff Problem (when does a Hausdorff space $X$ admit a bijective continuous mapping onto a Hausdorff compact space?) in the class of metric spaces of weight $\lambda=\lambda^{\aleph_0}$.