Suppose we have a directed system of inclusions of compact surfaces with boundary $$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \cdots $$ such that all of the surfaces $\{S_k\}$ have \emph{uniformly bounded topology}. That is, there exists some constant $C > 0$ independent of $k$ such that the genus and number of boundary components of $S_k$ is bounded above by $C$.

Then the surfaces have some well-defined direct limit surface $S$. I was wondering how pathological the topology of $S$ can become?

As of yet, I have been unable to produce any example that does not yield a punctured compact surface with finitely many punctures.

Venturing towards a possible proof of this statement, the uniform bound on the topology should imply via a quick Morse-theoretic argument that the surfaces $\{S_k\}$ become diffeomorphic for large $k$ and the inclusions are deformation retracts.

Suppose we have a directed system of inclusions of compact surfaces with boundary $$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \cdots $$ such that all of the surfaces $\{S_k\}$ have uniformly bounded topology. That is, there exists some constant $C > 0$ independent of $k$ such that the genus and number of boundary components of $S_k$ is bounded above by $C$.

Then the surfaces have some well-defined direct limit surface $S$. I was wondering how pathological the topology of $S$ can become?

As of yet, I have been unable to produce any example that does not yield a punctured compact surface with finitely many punctures.

Venturing towards a possible proof of this statement, the uniform bound on the topology should imply via a quick Morse-theoretic argument that the surfaces $\{S_k\}$ become diffeomorphic for large $k$ and the inclusions are deformation retracts.