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A [[Topological structure (topology)|topological structure (topology)]] on the space  $  D [ 0,1 ] $
of right-continuous functions on  $  [ 0,1 ] $
having limits to the left at each  $  t \in ( 0,1 ] $, 
introduced by A.V. Skorokhod [[#References|[a4]]] as an alternative to the topology of [[Uniform convergence|uniform convergence]] in order to study the [[Convergence in distribution|convergence in distribution]] of stochastic processes with jumps.

Let  $  \Lambda $
be the class of strictly increasing, continuous mappings of  $  [ 0,1 ] $
onto itself. For  $  \lambda \in \Lambda $
one defines

$$ 
\left \| \lambda \right \| =  \sup  _ {s \neq t } \left | { { \mathop{\rm log} } {
\frac{\lambda ( t ) - \lambda ( s ) }{t - s }
 } } \right | .
$$

The following distance, introduced by P. Billingsley [[#References|[a1]]], induces the Skorokhod topology and makes  $  D [ 0,1 ] $
a complete separable [[Metric space|metric space]]:

$$ 
d ( x,y ) =  \inf  _ {\lambda \in \Lambda } \left \{ \left \| \lambda \right \| +  \sup  _ {t \in [ 0,1 ] } \left | {x ( t ) - x ( \lambda ( t ) ) } \right | \right \} .
$$

An important property is that the Borel  $  \sigma $-
algebra associated with this topology coincides with the projection  $  \sigma $-
algebra.

The Skorokhod topology on the space  $  D [ 0, \infty ) $
of right-continuous functions on  $  [ 0, \infty ) $
having limits to the left can be defined by requiring the convergence in the Skorokhod metric on each compact interval  $  [ 0,T ] $, 
$  T > 0 $.

Applying Prokhorov's theorem [[#References|[a3]]] to the complete separable metric space  $  D [ 0, \infty ) $
yields that a sequence  $  \{ X _ {n} \} $
of  $  D [ 0, \infty ) $-
valued random variables (cf. [[Random variable|Random variable]]) converges in distribution if and only if their finite-dimensional distributions converge and the laws of  $  \{ X _ {n} \} $
are tight (for every  $  \epsilon > 0 $
there exists a compact set  $  K \in D [ 0, \infty ) $
such that  $  {\mathsf P} \{ X _ {n} \in K \} \geq  1 - \epsilon $
for all  $  n $). 
Useful criteria for weak convergence can be deduced from this result and from the characterization of compact sets in  $  D [ 0, \infty ) $(
see [[#References|[a1]]]).

Complete separable distances on the space  $  D ( T ) $
of functions with possible jumps on an arbitrary parameter set  $  T $
are introduced in [[#References|[a5]]], and for  $  T \subset  \mathbf R  ^ {k} $
these distances have been applied to obtain criteria for the convergence in law of multi-parameter stochastic processes.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Billingsley,   "Convergence of probability measures" , Wiley  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Pollard,   "Convergence of stochastic processes" , Springer  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y.V. Prokhorov,   "Convergence of random processes and limit theorems in probability theory"  ''Th. Probab. Appl.'' , '''1'''  (1956)  pp. 157–214</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.V. Skorokhod,   "Limit theorems for stochastic processes"  ''Th. Probab. Appl.'' , '''1'''  (1956)  pp. 261–290</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.L. Straf,   "Weak convergence of stochastic processes with several parameters" , ''Proc. Sixth Berkeley Symp. Math. Stat. and Prob.''  (1972)  pp. 187–221</TD></TR></table>