A universal continuum of weight $\aleph$

Abstract:

We prove that every continuum of weight $\aleph _1$ is a continuous image of the Čech-Stone-remainder $R^*$ of the real line. It follows that under $\mathsf {CH}$ the remainder of the half line $[0,\infty )$ is universal among the continua of weight $\mathfrak {c}$ — universal in the ‘mapping onto’ sense. We complement this result by showing that 1) under $\mathsf {MA}$ every continuum of weight less than $\mathfrak {c}$ is a continuous image of $R^*$, 2) in the Cohen model the long segment of length $\omega _2+1$ is not a continuous image of $R^*$, and 3) $\mathsf {PFA}$ implies that $I_u$ is not a continuous image of $R^*$, whenever $u$ is a $\mathfrak {c}$-saturated ultrafilter. We also show that a universal continuum can be gotten from a $\mathfrak {c}$-saturated ultrafilter on $\omega$, and that it is consistent that there is no universal continuum of weight $\mathfrak {c}$.