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Discrete Order Geometry (DOG) and Topology: A Preliminary Exploration of the Structured Description of Non‑Connected Spaces

Author: Zhang Suhang

Luoyang

Abstract

The establishment of Discrete Order Geometry (DOG) removes spatial connectivity as a necessary prerequisite for a geometric system, replacing it with hierarchical nested self‑similarity, order isomorphism, and continued fraction scale convergence as the core criteria for geometric determination. This paradigmatic shift is intrinsically consistent with the modern development of topology—from sensory connectivity to symbolic algebraic structures. As a perspective piece in the DOG series, this paper does not attempt to construct a fully fledged “DOG‑topology coupling system.” Instead, it raises several open questions and possible research directions: topological invariants for non‑connected spaces, the relationship between DOG hierarchical order and order topology, the feasibility of continued fraction convergent sequences as a topological approximation basis, and further applications of discrete topology within DOG’s empirical specimens. The aim is to invite scholars in topology, discrete mathematics, and geometry to explore this interdisciplinary direction.

Keywords: Discrete Order Geometry; DOG; topology; non‑connected space; order topology; topological invariant

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1 Introduction

The central breakthrough of Discrete Order Geometry (DOG) lies in removing spatial connectivity as an implicit prerequisite for a geometric system, and adopting hierarchical nested self‑similarity, order isomorphism, and continued fraction scale convergence as the criteria for determining the unity of a system. This line of thought subtly echoes the historical evolution of topology.

In traditional topology teaching and popular understanding, “connectivity” is often taken as an intuitive property of topological spaces—rubber‑sheet geometry, continuous deformation, no tearing or gluing. However, modern topology (algebraic topology, geometric topology, categorical topology) has long surpassed sensory intuition, moving toward symbolic algebraic structures: homology groups, homotopy groups, cohomology rings, spectral sequences, etc. Connectivity is merely one condition among many in most modern topological branches, far from the core. From this perspective, DOG’s advocacy that “non‑connected spaces can also have geometry” is not a challenge to topology, but rather a reflection of the same trend at a different level.

As a perspective piece in the DOG series, this paper does not attempt to construct a complete theoretical system, but rather raises several questions and directions that may link DOG and topology, with the aim of stimulating cross‑disciplinary discussion.

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2 The Position and Limitations of “Non‑Connectedness” in Current Topology

In point‑set topology, non‑connected spaces are usually treated as trivial cases: the discrete topology (all subsets open) and the indiscrete topology (only the empty set and the whole set open) are among the few non‑connected examples found in standard textbooks. While these examples satisfy the topological axioms, they lack structure—they have no intrinsic hierarchical order, no self‑similar nesting, no cross‑scale regularities.

In other words, topology can currently accommodate non‑connected spaces, but it has not yet systematically studied non‑connected spaces with a high degree of intrinsic order. The empirical specimens provided by DOG—the Sun‑Earth‑Moon system, the Galilean satellite resonance system—are natural examples of such “ordered non‑connected spaces.” Can topology provide invariants for these structures? This is the first open question.

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3 Open Question I: Topological Invariants for Non‑Connected Spaces

Classical topological invariants (number of connected components, fundamental group, homology groups) are usually defined and computed under the assumption of connectivity. For completely discrete spaces that nevertheless possess hierarchical nested order, can new invariants be defined?

Possible directions include:

· Hierarchical rank: based on the number of nesting levels in DOG, define the “order depth” of a space as an integer invariant.

· Self‑similar dimension: drawing on the concept of fractal dimension, but restricted to the spatial distribution and hierarchical ratios of discrete units, rather than continuous filling.

· Convergence order spectrum: treat the sequence of continued fraction convergents as a “spectrum” of spatial scales, i.e., an invariant as a sequence of rational numbers.

These invariants do not currently exist in standard topology, but they arise naturally from the structure of DOG.

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4 Open Question II: The Relationship Between DOG Hierarchical Order and Order Topology

Order topology is a branch of point‑set topology that endows partially ordered sets with a topological structure. One of the core features of DOG is precisely its hierarchical partial order—the Sun, Earth, and Moon form a strict subordination relation, and this relation repeats self‑similarly across nesting levels.

A natural conjecture: a DOG hierarchical order space can be viewed as an instance of some type of order topological space, where the partial order is defined by “orbiting” or “nested inclusion.” If this conjecture holds, then theorems from order topology on convergence, compactness, and connected components could be directly applied to DOG models, while DOG would provide macroscopic natural specimens for order topology.

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5 Open Question III: Continued Fraction Convergent Sequences as a Topological Approximation Basis

The continued fraction convergent sequence is the quantitative core of DOG. In topology, approximation bases or nets are among the fundamental tools for defining topological structures.

A question worth exploring is: can the continued fraction convergent sequence in DOG be regarded as a kind of “convergence basis” for a certain non‑connected space? More concretely, for a given irrational scale parameter (e.g., a period ratio), its continued fraction convergents form a sequence of rational numbers. Can we define a topology on a discrete set such that this sequence converges to the irrational limit, with the convergence order corresponding one‑to‑one to the DOG nesting levels? This might find analogies in p‑adic analysis or arithmetic topology.

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6 Open Question IV: Extending Discrete Topology Through Real‑World Instances

DOG has already presented the Sun‑Earth‑Moon system and the Galilean satellite system as macroscopic natural specimens for discrete topology. Further, what is the concrete manifestation of concepts in discrete topology—such as isolated points, boundary points, limit points—within these systems?

For example:

· In the DOG hierarchy, is each celestial unit an isolated point? How is its neighborhood defined (based on spatial distance vs. based on order hierarchy)?
· Does the “limit point” of the system correspond to the irrational limit of the continued fraction convergent sequence?
· What is the meaning of the “closure” operation in a hierarchical order space?

Addressing these questions would transform discrete topology from abstract definitions into an observable, computable, concrete theory.

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7 Concluding Remarks: An Interdisciplinary Direction Worth Exploring

The intersection between DOG and topology remains largely unexplored. This paper does not pretend—nor is it possible in a short perspective—to establish that intersection. It merely raises four open questions and invites scholars in topology, discrete mathematics, and geometry to explore them together:

1. How can topological invariants for ordered non‑connected spaces be defined?
2. What is the precise relationship between DOG hierarchical order and order topology?
3. Can continued fraction convergent sequences serve as a topological approximation basis for non‑connected spaces?
4. What are the concrete applications and extensions of discrete topology within DOG’s empirical specimens?

DOG provides a geometric paradigm and natural specimens; topology provides structural description and invariant theory. Their combination may give rise to a new research direction—neither purely symbolic nor tied to sensory connectivity—namely, the topology of discretely ordered spaces.

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References

[1] Zhang S H. Discrete Order Geometry (DOG): A New Geometric Paradigm Based on Fractal Nesting and Continued Fraction Scaling. 2026.
[2] Zhang S H. DOG Discrete Order Geometry and Its Coupling with Discrete Mathematics. 2026.
[3] Zhang S H. Discrete Order Geometry (DOG): A Study of Its Universal Application System. 2026.
[4] Zhang S H. Structural Shortcomings of Traditional Mathematical and Geometric Systems and the Systematic Complementation by the DOG Paradigm. 2026.
[5] Munkres J R. Topology[M]. Pearson, 2000.
[6] Kelley J L. General Topology[M]. Springer, 1955.
[7] Classical literature on order topology and partially ordered sets.

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