  Herein, a methodology is developed to replicate functions, measures and
stochastic processes onto a compact metric space. Many results are easily
established for the replica objects and then transferred back to the original
ones. Two problems are solved within to demonstrate the method: (1)
Finite-dimensional convergence for processes living on general topological
spaces. (2) New tightness and relative compactness criteria are given for the
Skorokhod space $D(\mathbf{R}^{+};E)$ with $E$ being a general Tychonoff space.
The methods herein are also used in companion papers to establish the: (3)
existence of, uniqueness of and convergence to martingale problem solutions,
(4) classical Fujisaki-Kallianpur-Kunita and Duncan-Mortensen-Zakai filtering
equations and stationary filters, (5) finite-dimensional convergence to
stationary signal-filter pairs, (6) invariant measures of Markov processes, and
(7) Ray-Knight theory all in general settings.
