  A channel $W$ is said to be input-degraded from another channel $W'$ if $W$
can be simulated from $W'$ by randomization at the input. We provide a
necessary and sufficient condition for a channel to be input-degraded from
another one. We show that any decoder that is good for $W'$ is also good for
$W$. We provide two characterizations for input-degradedness, one of which is
similar to the Blackwell-Sherman-Stein theorem. We say that two channels are
input-equivalent if they are input-degraded from each other. We study the
topologies that can be constructed on the space of input-equivalent channels,
and we investigate their properties. Moreover, we study the continuity of
several channel parameters and operations under these topologies.
