We show that a Tychonoff space is the perfect pre-image of a cofinally complete metric space if and only if it is paracompact and cofinally Čech complete. Further properties of these spaces are discussed. In particular, cofinal Čech completeness is preserved both by perfect mappings and by continuous open mappings.

If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a metric extension of $X$. If $T_1$ and $T_2$ are metric extensions of $X$ and there is a continuous map of $T_2$ into $T_1$ keeping $X$ pointwise fixed, we write $T_1\le T_2$. If $X$ is noncompact and metrizable, then $\left(ℳ\right(X),\le )$ denotes the set of metric extensions of $X$, where $T_1$ and $T_2$ are identified if $T_1\le T_2$ and $T_2\le T_1$, i.e., if there is a homeomorphism of $T_1$ onto $T_2$ keeping $X$ pointwise fixed. $\left(ℳ\right(X),\le )$ is a large complicated poset studied extensively by V. Bel’nov [The structure of...