The topology of a closed surface can be constructed by identifying edges of a 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!

The topology of a closed surface can be constructed by identifying edges of a 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.