Applications for set theory, ordinal theory, infinite combinatorics and general topology in computer science? I am a mathematician interested in set theory, ordinal theory, infinite combinatorics and general topology.

Are there any applications for these subjects in computer science?
I have looked a bit, and found a lot of applications (of course) for finite graph theory, finite topology, low dimensional topology, geometric topology etc.

However, I am looking for applications of the infinite objects of these subjects, i.e. infinite trees (Aronszajn trees for example), infinite topology etc.

Any ideas?

Thank you!!
One major application of topology in semantics is the topological approach to computability. 

The basic idea of the topology of computability comes from the observation that termination and nontermination are not symmetric. It is possible to observe whether a black-box program terminates (simply wait long enough), but it's not possible to observe whether it doesn't terminate (since you can never be certain you have not waited long enough to see it terminate). This corresponds to equipping the two point set {HALT, LOOP} with the Sierpinski topology, where $\emptyset, \{HALT\}, and \{HALT, LOOP\}$ are the open sets. So then we can basically get pretty far equating "open set" with "computable property". One surprise of this approach to traditional topologists is the central role that non-Hausdorff spaces play. This is because you can basically make the following identifications

$$
\begin{matrix}
\mathbf{Computability} &amp; \mathbf{Topology}\\
\mbox{Type} &amp; \mbox{Space} \\
\mbox{Computable function} &amp; \mbox{Continuous function} \\
\mbox{Decidable set} &amp; \mbox{Clopen set} \\
\mbox{Semi-decidable set} &amp; \mbox{Open set} \\
\mbox{Set with semidecidable complement} &amp; \mbox{Closed set} \\
\mbox{Set with decidable equality} &amp; \mbox{Discrete space} \\
\mbox{Set with semidecidable equality} &amp; \mbox{Hausdorff space} \\
\mbox{Exhaustively searchable set} &amp; \mbox{Compact space} \\
\end{matrix}
$$

Two good surveys of these ideas are MB Smyth's Topology in the Handbook of Logic in Computer Science and Martin Escardo's Synthetic topology of data types and classical spaces.

Topological methods also play an important role in the semantics of concurrency, but I know much less about that.