  Given a general source $\sV=\{V^n\}\noi$ with {\em countably infinite} source
alphabet and a general channel $\sW=\{W^n\}\noi$ with arbitrary {\em abstract}
channel input and output alphabets, we study the joint source-channel coding
problem from the information-spectrum point of view. First, we generalize
Feinstein's lemma (direct part) and Verd\'u-Han's lemma (converse part) so as
to be applicable to the general joint source-channel coding problem. Based on
these lemmas, we establish a sufficient condition as well as a necessary
condition for the source $\sV$ to be reliably transmissible over the channel
$\sW$ with asymptotically vanishing probability of error. It is shown that our
sufficient condition coincides with the sufficient condition derived by Vembu,
Verd\'u and Steinberg, whereas our necessary condition is much stronger than
the necessary condition derived by them. Actually, our necessary condition
coincide with our sufficient condition if we disregard some asymptotically
vanishing terms appearing in those conditions. Also, it is shown that {\em
Separation Theorem} in the generalized sense always holds. In addition, we
demonstrate a sufficient condition as well as a necessary condition for the
$\vep$-transmissibility ($0\le \vep <1$). Finally, the separation theorem of
the traditional standard form is shown to hold for the class of sources and
channels that satisfy the (semi-) strong converse property.
