A semilattice of degrees of computable metrics

Under study is the ordering $\mathcal{{CM}}_c(\mathbf{X})$ of $c$-degrees of computable metrics on a Polish space $\mathbf{X}$ with a distinguished dense subset. We prove that this ordering forms a lower semilattice. If, for a computable metric $\rho$ on $\mathbf{X}$, there is a computable limit point in $(X,\rho)$; it is possible to construct a computable metric $\rho'<_c\rho$. Under the same assumption, there exists a computable metric $\widehat{\rho}$ such that $\deg_c(\rho)$ and $\deg_c(\widehat{\rho})$ have no common upper bounds in $\mathcal{{CM}}_c(\mathbf{X})$; thus, in this case $\mathcal{{CM}}_c(\mathbf{X})$ is neither an updirected poset nor an upper semilattice.