  In this paper, the Teichm{\"u}ller spaces of surfaces appear from two points
of views: the conformal category and the hyperbolic category. In contrast to
the case of surfaces of topologically finite type, the Teichm{\"u}ller spaces
associated to surfaces of topologically infinite type depend on the choice of a
base structure. In the setting of surfaces of infinite type, the
Teichm{\"u}ller spaces can be endowed with different distance functions such as
the length-spectrum distance, the bi-Lipschitz distance, the Fenchel-Nielsen
distance, the Teichm{\"u}ller distance and there are other distance functions.
Unlike the case of surfaces of topologically finite type, these distance
functions are not equivalent. We introduce the finitely supported
Teichm{\"u}ller space T f s H 0 associated to a base hyperbolic structure H 0
on a surface $\Sigma$, provide its characterization by Fenchel-Nielsen
coordinates and study its relation to the other Teichm{\"u}ller spaces. This
paper also involves a study of the Teichm{\"u}ller space T 0 ls H 0 of
asymptotically isometric hyperbolic structures and its Fenchel-Nielsen
parameterization. We show that T f s H 0 is dense in T 0 ls H 0 , where both
spaces are considered to be subspaces of the length-spectrum Teichm{\"u}ller
space T ls H 0. Another result we present here is that asymptotically
length-spectrum bounded Teichm{\"u}ller space A T ls H 0 is contractible. We
also prove that if the base surface admits short curves then the orbit of every
finitely supported hyperbolic surface is non-discrete under the action of the
finitely supported mapping class group MCG f s $\Sigma$ .
