The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper {\em radical} ideals (for example, $\{0\})$ that are closed under the natural metric, but has no {\em prime} ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J.\,Gleason. From this example, rings of arbitrary characteristic with the same properties are obtained. The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically {\em either} the inequality $d(x\vee y,\,x\vee z)\leq d(y,z)$ {\em or} the inequality $d(x\wedge y,\linebreak[2]\,x\wedge z)\leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; that is, $L$ is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,\,x\vee y)\leq d(x,y),$ respectively $d(x,\linebreak[2]\,x\wedge y)\linebreak[2]\leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.

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