There's an interesting branch of math called "synthetic topology" where you study computational questions topologically. https://ncatlab.org/nlab/show/synthetic

There's an interesting branch of math called "synthetic topology" where you study computational questions topologically. https://ncatlab.org/nlab/show/synthetic+topology https://ncatlab.org/nlab/show/synthetic+topology Given a set X of things that can be encoded for a Turing machine, consider the semidecidable subsets (those subsets of things for which a there's a Turing machine that will halt if and only if it's given one of them). This forms a basis for a topology; it has finite intersections because you can run two Turing machines in parallel. So, a subset S of X is considered to be open if every element p in S has a Turing machine that halts for p, and whenever it halts for any other element, that element is in S. Given this topology, what does it mean to be compact? It means whenever you have a collection of Turing machines (possibly infinite!) such that for every element of X there's a Turing machine that halts for it, then there's some finite sub-collection of Turing machines with the same property. The space of binary streams can be encoded as plain old Turing machine tapes (maybe give it to the Turing machine as a second tape for convenience). If a Turing machine halts given a tape, then it only looked at some prefix of the stream to make the decision, so it will also halt if you change anything outside that prefix. This corresponds to the usual product topology, with the basic Turing machines being "check that the stream has this fixed prefix." The space is a product of finite sets, so by the Tychonoff theorem it's compact. An application of this is the following. Suppose you have a computable function f : binary streams -> bool. That is, you have a Turing machine that halts for every binary stream, resulting in either True or False. You can think of this as being two Turing machines, one that recognizes when f returns True (otherwise it artificially goes into an infinite loop), and similarly another that recognizes when f returns False (otherwise loops). Both of these Turing machines define open subsets of the space of binary streams, the union of which is the whole space of binary streams. Since they are open, they are all unions of all the basis sets of the product topology (recall: sets of all binary sequences with a common prefix). Since the space is compact, you only need finitely many of these basis sets -- let n be the longest prefix. Hence, there is a Turing machine for f that makes its decision by looking at only a prefix of a fixed length n of the binary stream!