The equivalences to and the connections between the modulus-of-continuity condition, compact containment and tightness on
D
E
[
a
,
b
]
D_{E}[a,b]
with
a
>
b
a>b
are studied. The results within are tools for establishing tightness for probability measures on
D
E
[
a
,
b
]
D_E[a,b]
that generalize and simplify prevailing results in the cases that
E
E
is a metric space, nuclear space dual or, more generally, a completely regular topological space. Applications include establishing weak convergence to martingale problems, the long-time typical behavior of nonlinear filters and particle approximation of cadlag probability-measure-valued processes. This particle approximation is studied herein, where the distribution of the particles is the underlying measure-valued process at an arbitrarily fine discrete mesh of points.