The topology of a closed surface can be constructed
by identifying edges of a [fundamental polygon](http://en.wikipedia.org/wiki/Fundamental_polygon) 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:


1. 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 $n \rightarrow \infty$:
$$a\_1 a\_1 a\_2 a\_2 a\_3 a\_3 \cdots a\_i a\_i \cdots \;.$$
2. 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:
$$.011100100011\ldots \leftrightarrow
 .100011011100\ldots \;.$$


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!


**Edit.** What was nonsensical was my bit-complement example, as pointed out by both
Victor and Sergei. I'll leave it so their remarks make sense. I intended a more patternless
pairing. 



Fundamental polygons with infinite pairwise identifications
Identification spaces with infinitely complicated identifications do occur naturally in various contexts. Here are a few more remarks:


1. If you define an involution on a countable set of intervals on the boundary of a surface, as in Victor Protsak's remark, and extend the equivalence relation to finite equivalence classes of the closure (i.e., vertices), you can construct arbitrary noncompact surfaces. The classification of these is known, and rather interesting. In the orientable case: The first invariant is the set of *ends* of the space, meaning, the inverse limit of the set of components of complements of compact sets. I think this is the same as the closure of the equivalence relation extended to the rest of the circle, minus the surface, but I haven't thought it through properly. The set of ends can be an arbitrary closed homemorphism type of compact totally disconnected space. I believe it's known that the number of types has the cardinality of the continuum; it's easy to see it's at least $\omega$ and at most $2^\omega$. The second invariant is a closed subset of the set of ends, that is the limit of "handles", that is, pairs of curves that intersect in one point. Finally, if there are only finitely many handles, you need the genus.
2. There is a structure called a geodesic lamination in the disk: a closed collection of disjoint hyperbolic lines ending at infinity. Given a lamination, you can identify the endpoints. This kind of identification arises in multiple ways. First, it describes the structure of the cut locus" for a curve in the plane. For which pairs of points is there a circle whose interior is contained on the inside, but touches only those two points? Second, if you identify all lines to points, the resulting space is still homeomorphic to the plane. They give explanations for the structure of Julia sets and limit sets of Kleinian groups, etc.
More generally, one can represent identifications by collections of lines that are allowed to cross, but the theory becomes more complicated. In the case of Julia sets, the identification is in some sense isometric. (Actually, the involution is nontrivial only on a set of measure 0 with dense complement, but it is a limit of isometric involution on finite collections of intervals.) If you identify the circle using two geodesic laminations, one on the inside and one on the outside, you typically get $S^2$, and the circle becomes a space-filing curve: a sort of Hamiltonian path that is a limit of simple curves.
3. For a dynamical system, it is natural to look at the set of all orbits. This is an identification space that is somtimes "nice" but more often is (in Ivanov's words) weird: non-Hausdorff and failing other separation properties. Nonetheless, these quotient spaces often carry structure that is helpful to understanding the dynamics. A simple example: the flow in the plane minus the origin $\phi\_t(x,y) = (\exp(t) x, \exp(-t) y) of a linear differential equation. The orbit space is a 1-dimensional manifold, because you can take a cross-section near any point that meets any orbit at most once, so it embeds in the quotient space. However, the quotient is not Hausdorff.


