On boundedly compact metrics and UC metrics

Let $P_1$ and $P_2$ be potential properties of a metric for which each of $P_1 \wedge P_2, P_1 \wedge \neg P_2, \neg P_1 \wedge P_2$ and $\neg P_1 \wedge \neg P_2$ is possible. We call such a pair \textit{strongly independent} if whenever a metrizable space admits a metric for which either $P_1$ or $\neg P_1$ holds and separately a metric for which either $P_2$ or $\neg P_2$ holds, then there must exist a compatible metric for which both conditions at once hold. We show that boundedly compactness and UC-ness form a strongly independent pair and so do boundedness and UC-ness. Metrizable spaces that admit a boundedly compact metric when equipped with a non-UC metric have been recently studied with respect to the lineability of the non-uniformly continuous real-valued functions defined on them within $C(X)$ [4].