Computable planar paths intersect in a computable point

TL;DR

This paper proves that for two computable paths in the unit square with specified endpoints, their intersection point is also computable, extending classical existence results to the realm of computability theory.

Contribution

It establishes that the intersection point of two computable paths with given endpoints in the unit square is itself computable, bridging topology and computability.

Findings

Proves computability of intersection point for computable paths

Extends classical intersection existence to computable analysis

Shows computability results for paths with fixed endpoints

Abstract

Consider two paths $f,g:[0;1]\to [0;1]^2$ on the unit square such that $f(0)=(0,0)$, $f(1)=(1,1)$, $g(0)=(0,1)$, $g(1)=(1,0)$, $f(0;1)\subseteq (0;1)^2$ and $g(0;1)\subseteq (0;1)^2$. By continuity of $f$ and $g$ there is a point of intersection. We prove that there is a computable point of intersection if the paths are computable.