The Existence of the Halting Function in a Purely Mathematical Context and the Solution of Halting paradox with termination checking of programs over definite Topological spaces. (AMS MSC2020 03-08) Category: Mathematics and computation

Abstract: Here we define, mathematically, a program f_(i ):w⟶〖{0,1}〗_(ℵ_0 ). Where w is a set of all programmable words, we consider as the domain, and 〖{0,1}〗_(ℵ_0 )is the co-domain is the set of all finite or infinite strings of 0 & 1. (*Ref.1). In this paper, we propose a function f_(i )*, which we call the stop function and we propose another function h, which we call the halt function. Our objective of the paper is to show their existence in a completely mathematical form of the, well known halting problem and its solution using simple functional compositions. Our next approach is to study the structure of the domain of programmable functions i.e. w and its topology with respect to the topology of 〖{0,1}〗_(ℵ_0 ) . Followed by defining the finite string topology and the product topology on 〖{0,1}〗_(ℵ_0 )and study the continuous functions from w to 〖{0,1}〗_(ℵ_0 ). Our main intention is to show that a programmable function will terminate for a specific input if and only if the function is continuous at that specific input (point on w). The main deference between the classical halting problem and our method is instead of mechanical switch program we utilize two-point functions. Which leads us to generate some interesting results through a purely mathematical context.

Keywords: Halting Problem, Turing machine, Undecidability, Stop function, Halting function,Product topology