What question did this study set out to answer?

The study aims to prove that topology serves as a fundamental connection between discrete and continuous structures in mathematics.

June 22, 2026Open Access

Topology as a Bridge: The Unification of the Discrete and the Continuous

Key Points

The study aims to prove that topology serves as a fundamental connection between discrete and continuous structures in mathematics.
Developed the 'Topology-as-Bridge' theory based on the information pixel axiom system.
Proved a three-stage transition theorem from discrete structures to continuous functions, integrating key axioms for a rigorous framework.
Utilized examples like dynamic number theory to illustrate the discrete-continuous connection.
Established that discrete information pixels form topological structures through phase coupling.
Demonstrated that topology is essential for the transition from discrete to continuous systems, showing emergent continuity in macroscopic behavior.
Proved that topology is the minimal necessary structure for connecting discrete and continuous information.

Abstract

Abstract This paper proposes the "Topology-as-Bridge" (TaB) theory, rigorously proving that within the information pixel axiom system, topological structure is not an appendage of geometry, but a natural bridge connecting the discrete and the continuous, intrinsically emerging from the local coupling of Axiom 4. The core discovery: discrete information pixels form neighborhood relations through the phase coupling of Axiom 4; these neighborhood relations satisfy the three open-set axioms of a topological space, naturally constituting a topological structure; this topology, under the joint action of the closure condition of Axiom 4 and the finiteness of Axiom 2, necessarily converges to a stable topology; the uniformization of the phase distribution on the stable topology directly produces emergent continuity—its macroscopic effective behavior can be described by traditional continuous functions. This paper rigorously proves a three-stage transition theorem—discrete to topology (Theorem 2.1), topology to convergence (Theorem 2.3), convergence to continuous (Theorem 2.4)—and integrates them into the Topology Bridge Theorem (Theorem 2.5), completely establishing a rigorous mathematical channel from the discrete microscopic to the continuous macroscopic. The entire argument directly anchors all theorems on Axioms 1–4, upgrading Axiom 5 (the Correlation Emergence Principle) from an axiom to a rigorously derivable theorem. This paper further points out that the root of the discrete-continuous split in classical mathematics lies in the systematic absence of phase information from mathematical objects—generativism fills this blind spot through the phase coupling of Axiom 4, making the topology bridge possible. Taking dynamic number theory as an example, it demonstrates the three-stage transition between the discrete periodic spectrum, the topological phase closure, and the continuous rotating vector. This paper further rigorously proves: within the information pixel axiom system, topology is the unique bridge connecting the discrete and the continuous---any other possible transition structure contains topology, and topology is the minimal necessary structure of information connection.This paper and The Pathology of the Continuum and the Discrete Substrate form a mirror binary star—one proving from the negative side that the discrete cannot emerge from the continuous, the other proving from the positive side how the continuous emerges from the discrete—together constituting the metamathematical foundation of generative mathematics. Keywords: topology as a bridge; discrete and continuous unification; emergent continuity; Axiom~4; phase coupling; local coupling; phase closure; isoperimetric optimality; metamathematical foundation; Dedekind cuts; phase blind spot; dynamic number theory

Topology as a Bridge: The Unification of the Discrete and the Continuous

Key Points

Abstract

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