| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies topological spaces defined by open sets, continuity, convergence, compactness, connectedness, separation axioms, and product constructions. Excludes algebraic or geometric structures unless explicitly imposed. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at all structural scales: from local neighborhoods and bases at points to global properties like compactness, connectedness, metrizability, and product topologies. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Sets, topologies, open/closed sets, bases, subbases, continuous maps, nets, filters, closures, interiors, products, quotient spaces. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Openness, closedness, compactness, connectedness, Hausdorff property, regularity, normality, completeness (in metric contexts), convergence behavior of nets/filters. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Topological spaces, metric spaces, Hausdorff spaces, compact spaces, connected spaces, product spaces, quotient spaces, separable spaces, first/second countable spaces. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Choice of topology, neighborhood bases, convergence parameters, filter or net selections, cardinalities, separation levels, compactness indicators. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded by bases/subbases, convergence structures (nets/filters), open-cover definitions of compactness, separation axioms, and product/quotient constructions. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Assuming nicer topologies (e.g., Hausdorff, compact, metrizable); using countable bases; restricting attention to metric spaces; ignoring pathological examples unless required for generality. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down for non-Hausdorff spaces, non-regular spaces, wild quotients, non-first-countable spaces, or spaces lacking standard convergence behavior. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes set-theoretic foundations (usually ZFC); classical definitions of open sets; convergence via nets/filters; stability of product constructions; predictable behavior of closures/interiors. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes topology captures “shape” at all scales; convergence via nets/filters is comprehensive; axioms (T0–T4) provide meaningful classification; product and quotient operations behave systematically. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires compatibility of open sets, continuity definitions, convergence rules, and separation axioms; constructions like products and quotients must preserve topological coherence. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires alignment between bases, continuity rules, closure/interior operators, compactness and convergence criteria, and categorical behavior under maps and constructions. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Convergence (sequences, nets, filters), continuity behavior, compactness via open covers, connectedness patterns, separation-axiom effects, product and quotient behavior. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Sequences fail in non-first-countable spaces; nets/filters required for general convergence; pure topology cannot detect geometric/metric structure; some separation failures undetectable by sequences alone. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Cardinality of bases, size of open covers, length of refinement chains, T0–T4 separation level, compactness via finite subcovers. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Bases, subbases, nets, filters, closure and interior operators, open-cover systems, product and quotient constructions. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of continuity, compactness, connectedness, closure, interior, convergence via nets/filters, separation axioms, product and quotient topology. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Testing continuity via preimages of opens; constructing bases; verifying compactness via subcovers; testing connectedness by attempted separation; applying closure/interior operators. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Forming open covers; generating subbases; building product topologies; constructing quotient spaces; producing nets/filters for convergence inspection. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling neighborhoods; selecting nets/filters in non-metrizable spaces; sampling covers to test compactness; sampling connected subsets; sampling identifications in quotient spaces. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Collections of open sets, closure/interior tables, convergence diagrams (nets/filters), compactness cover-data, separation-axiom classification tables, product/quotient diagrams. |
| | Resolution | The granularity or precision with which data is captured. | Resolution depends on refinement of bases, granularity of covers, strength of nets/filters, precision of separation distinctions, and fidelity of product/quotient constructions. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Checking base/subbase correctness; validating continuity tests; verifying compactness via proper subcovers; confirming closure/interior consistency; ensuring independence from base choice. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Misidentified open sets; incorrect convergence in non-first-countable spaces; failure to detect non-compactness; incorrect product or quotient topology; misclassification of separation properties. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Closure/interior duality, continuity via preimages of open sets, compactness via finite subcovers, connectedness via absence of separation, stability of product topologies, quotient-topology inheritance patterns. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Connectedness, compactness, separation levels (T0–T4), cardinal invariants (weight, density, character), convergence classes, topological equivalence under homeomorphism. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Convergence mechanisms via nets/filters; open-set mechanisms defining continuity; closure operators generating limit points; product/quotient mechanisms shaping global topology; compactness enforced by open-cover mechanisms. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Refinement of bases; construction of product spaces; quotient identification pathways; convergence-pathways of nets; closure-iteration pathways; open-cover refinement sequences. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Topological space, open/closed sets, basis, subbasis, continuity, compactness, connectedness, nets, filters, separation axioms, product topology, quotient topology. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | T0–T4 spaces, compact vs. non-compact, connected vs. disconnected, first/second-countable, separable, metrizable, locally compact, Lindelöf spaces. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Closure/interior operators, continuity condition (f^{-1}(U)), compactness via finite subcovers, definitions of product topology, quotient-map condition, filter/nets convergence relations. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Standard R^n with usual topology, discrete/indiscrete spaces, product spaces, quotient spaces (identifications), Sierpiński space, cofinite topology, metric-induced topologies. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Discrete spaces, simple quotient spaces, metric spaces as special cases, finite topologies, canonical non-Hausdorff examples, standard compact spaces (e.g., Cantor set). |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdown of sequence-based convergence in non-first-countable spaces; product topology pathologies; failures of normality; noncompactness; quotient spaces losing Hausdorff property; metrizability obstructions. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Set-theoretic topology, product/quotient theory, compactness theory, convergence theory, metrization theorems, category-theoretic viewpoint of Top. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to analysis (continuity/compactness foundations), geometry (metric spaces), algebra (topological groups), logic (set theory, cardinal invariants), computer science (domain theory, semantics). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying topologies on a set; modifying bases/subbases; altering product or quotient constructions; adjusting convergence structures (nets vs. filters) to test compactness, continuity, and separation properties. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural behavior of continuity, convergence, compactness, separation, and connectedness in fixed topologies without modifying the underlying space or base structure. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing continuity via preimage-open sets; verifying compactness using open covers; checking connectedness via attempted separation; evaluating separation axioms; testing convergence through nets/filters. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Repeating continuity tests with different bases; recomputing compactness with alternate open covers; re-evaluating convergence with nets vs. filters; re-testing separation properties under equivalent bases. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: analyzing frequency of compactness failures, comparing convergence under different subnet selections, evaluating stability of separation axioms, examining refinement-structure behavior. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing topologies on the same set; comparing product vs. quotient behavior; evaluating metrizability criteria; comparing separation levels; comparing convergence behavior across different structures. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting misidentified open sets; incorrect continuity tests; false compactness conclusions; misapplied closure/interior operators; convergence errors in non-first-countable spaces; quotient misidentification. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding bias toward metric examples; preventing reliance on sequence-based reasoning where nets are required; ensuring bases/subbases chosen do not artificially simplify the topology; avoiding selective open covers that force compactness. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Independent verification of continuity, compactness, separation, and convergence proofs; reviewing quotient and product constructions; validating open-cover arguments. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Refining topological definitions; modifying constructions for product or quotient spaces; updating metrizability or separation criteria; revising convergence frameworks when counterexamples arise. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Fully disclosing choice of bases/subbases, open-cover selections, convergence method (sequences vs. nets vs. filters), and assumptions in product/quotient constructions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Accurate reporting of counterexamples; honest treatment of non-metrizable phenomena; avoidance of sequence-based shortcuts; proper attribution of topological theorems, lemmas, and constructions. |