Structure-Sensitive Mathematics IV: Omega-Topology and Omega-Geometry — Structural Proximity, Geodesics, and Geometric Consistency

This work presents the fourth paper of the Structure-Sensitive Mathematics (SSM) research series, extending the Omega framework into topology, geometry, and structural consistency analysis.

Building upon Omega-sensitive function spaces, Omega-Banach/Hilbert structures, and Omega-operator theory developed in Parts I–III, this study investigates how structural information can become part of mathematical notions of proximity, distance, and geometry.

The central extension is the Omega-geometric representation:

MΩ = (M, g, Ω)

with the Omega metric:

dΩ(x,y) = dg(x,y) + λ|Ω(x) − Ω(y)|

where classical geometric distance is combined with structural difference.

The paper introduces and explores:

• Omega-topological spaces  
• Omega-open sets  
• Omega-convergence in topology  
• Structural discontinuity points  
• Omega-continuous maps  
• Omega-manifolds  
• Omega-metric geometry  
• Omega-geodesic selection principle  
• Omega-curvature concepts  
• State-dependent Omega topology  
• Connections with the Law of Geometric Consistency (LGC)

The central question explored is:

Can two states be geometrically close while being structurally different?

The proposed framework extends mathematical geometry from:

positional proximity

toward:

positional + structural proximity.

Potential research connections include:

• topology  
• differential geometry  
• information geometry  
• complex systems  
• optimization  
• quantum information  
• artificial intelligence reliability  
• structure-aware physical modeling  

This work does not replace classical topology or Riemannian geometry, but explores an additional structural layer for analyzing hidden divergence, structural compatibility, and geometry-dependent evolution.

Part IV of V — Foundations of Omega-Based Structural Analysis.