There seems to be an analogy between semidecidability and topology, but to me, the "topology" seems to be weak. The usual definition of a topology over a space $X$ is:
Statement #1: $\emptyset$ and $X$ are open.
Statement #2: Union of arbitrarily many open sets is open.
Statement #3: Intersection of finitely many open sets is open.
But the "topology" in analogy to semidecidability weakens Statement #2 to:
- Union of a computable sequence of open sets is open.
For example, give $\mathbb{N}$ the "topology" analogous to semidecidability. Then every singleton set is open (in fact, clopen). Let $H$ be the halting set. Then $H$ is open. But $\mathbb{N} \setminus H$ isn't open, despite being a countable union of singleton sets.
This means the usual basis (the singleton sets) doesn't "generate" the discrete topology.
I have so many questions, such as the impact on hierarchy of separation axioms, but for now, let me focus on one question. Is there a notion of "basis" for semidecidability? If so, how does it generate the "topology"?
