The topology of a closed surface can be constructed
by identifying edges of a [fundamental polygon][1] of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
the construction can be specified by a string of $2n$ symbols:
$a b a^{-1} b^{-1}$ for the torus, $a a b b$ for the Klein bottle, etc.

My question is: Does it make sense to have an infinite
number of boundary identifications, i.e., does it
define some topological object?

More specifically:

<ol>
<li> Does an infinite string of symbols representing pairwise
identifications correspond to some surface?
For example, the generalization of a non-orientable genus-$n$ surface as $n \rightarrow \infty$:
$$a_1 a_1 a_2 a_2 a_3 a_3 \cdots a_i a_i \cdots \;.$$
</li>

<li> Does it make sense to have an uncountable number of pairwise point identification
around a circle?  
For example, parametrize the circle circumference from 0 to 1
and identify points with complementary binary representations:
$$.011100100011\ldots  \leftrightarrow
 .100011011100\ldots \;.$$
</li>
</ol>

These extensions may be nonsensical, in which case I apologize for the distraction!
But if something along these lines has been studied, I'd appreciate a reference.  Thanks!

  [1]: http://en.wikipedia.org/wiki/Fundamental_polygon