Compactness lab
The compactness equivalence web
Compactness has several faces: covers, closed-set intersections, nets, filters, sequences, total boundedness, completion, and the Euclidean closed-and-bounded theorem. The graph separates true equivalences from nearby compactness-like properties, so the proof routes show both the shortcuts and the traps.
Greedy proof lab
Minimum spanning tree proof web
MST theory is a compact little universe: slice/cut and cycle/circle properties, exchange arguments, Prim and Kruskal invariants, minimax paths, threshold connectivity, lexicographic bottleneck optimality, matroid greedy, and nearby statements that look similar but are weaker or stronger. This panel treats those statements as proof faces and lets the bridges form a clique where that is valid.
Fourier proof lab
Spatial and frequency operator zoo
Convolution, multiplication, differentiation, Laplacians, Green functions, heat flow, deconvolution, sampling, and windowing form a beautiful dictionary. The panel shows which identities are true dualities, which are recipe routes, and which are warnings about instability or discretization.
Trig identity lab
Trigonometry, complex numbers, and hyperbolic functions
The same small engine appears as circle geometry, complex multiplication, power series, rotation matrices, and hyperbolic motion. This zoo keeps the definitions, proof routes, and caveats visible at once: Euler formula, cis, angle addition, De Moivre, sinh/cosh, boosts, analytic continuation, and the branch-cut warning for logarithms.
Fractal geometry lab
Dimension definitions and examples
Fractal geometry has its own compact zoo of dimensions: Hausdorff, box/Minkowski, packing, Assouad, similarity, Rényi, correlation, information, spectral, walk, and multifractal viewpoints. This panel separates equalities that need self-similar hypotheses from inequalities, examples, and warnings where two dimension ideas are answering different questions.
Noise and roughness lab
fBM, spectra, and procedural noise
Noise is a bridge between probability, Fourier analysis, fractal geometry, and graphics. This zoo separates stochastic processes such as Brownian motion and fractional Brownian motion from colored-noise spectra such as red and blue noise, plus procedural recipes such as value, Perlin, simplex, octave, turbulence, and ridged noise.
Spectral lab
Spectral & eigendecomposition
A matrix's spectrum is described by a web of definitions and theorems that prove one another — but only some links run both ways. Here we separate genuine equivalences from one-way implications, scope facts, and tempting false friends.
Curvature lab
Curvature & Gauss-Bonnet
The curvature cluster: how the two fundamental forms generate every local curvature notion, why Gauss's curvature is secretly intrinsic, and how local turning sums to a global topological invariant. Watch which links are genuine equivalences and which only run one way.
Choice lab
The Axiom of Choice and its equivalents
Over ZF, a surprising web of theorems from across mathematics all turn out to be the same statement in disguise. The point of this zoo is to separate the genuine full-strength equivalents from the strictly weaker fragments that merely follow from Choice.
Choice lab
Induction principles
A cluster of induction schemes that look different but mostly prove the same thing. Over the Peano axioms, weak, strong, and well-ordering on ℕ are the same principle in three costumes; the rest are either special cases of well-founded induction or its generalization to well-ordered domains.
Choice lab
The complexity class zoo
Most class relations are proven one-way inclusions; only a handful are true equivalences, and the most famous questions remain stubbornly open. The art is telling these apart.
Field theory · Galois theory
The Galois Correspondence
A cluster of definitions and theorems where equivalent characterizations prove one another. The keystone is the Fundamental Theorem of Galois Theory: for a finite normal-and-separable extension, subgroups of the Galois group correspond, inclusion-reversingly, to intermediate fields. Around it sit the genuine equivalences (Galois ⇔ normal+separable; solvable-by-radicals ⇔ solvable group; constructible ⇔ tower of quadratics), the one-way consequences (finite fields ⇒ cyclic Galois group), and the false friends (degree a power of 2 is necessary, not sufficient, for constructibility).
Abstract algebra
Group theory core
The objects, maps, and counting theorems that organize symmetry. The spine is normality: a subgroup is normal exactly when it is a kernel, exactly when the quotient is a group — three faces of one idea. Around it sit the structural theorems (Lagrange, Sylow, Cayley, orbit-stabilizer) and the special classes (cyclic, abelian, simple). Watch the classic false friends: Lagrange has no converse, and normality is not transitive.
Measure theory
Measure-theory foundations
The scaffolding of integration: which sets you can measure (sigma-algebras, Borel, Lebesgue), how you assign size (measures, outer measures, the Caratheodory extension), which functions you can integrate (measurable functions, almost everywhere), and the structural theorems (Radon-Nikodym, Fubini-Tonelli). True equivalences are separated from one-way consequences, conditional scope (sigma-finiteness, completeness), and false friends (Borel vs Lebesgue, a.e.-equal vs equal).
Convergence lab
Modes of convergence
A sequence of functions (or random variables) can converge in many inequivalent senses. The web below sorts strict one-way implications from the conditional bridges (finite measure, subsequences, uniform integrability) and the famous counterexamples that keep the arrows from reversing.
Integral zoo
Which functions become integrable?
Integral definitions form a small hierarchy of scopes: bounded interval sums, measure-theoretic integration, improper limits, Stieltjes weights, gauge refinements, and weak testing against smooth probes. The key question is not just “what is the value?” but “which functions are admitted, and under which hypotheses?”
Separation ladder
From T0 to the full separation taxonomy
This expanded module follows the named axioms on Wikipedia's separation-axiom page: the main definitions, the summary-table additions, and the “other separation axioms” section. Switch between the connection map and focused axiom families, then pick any axiom to see the witness condition appear on an interactive diagram.
Counterexample lab
Spaces that separate similar definitions
Inspired by Steen and Seebach's Counterexamples in Topology and modern searchable counterexample databases, this panel collects small spaces whose job is to say exactly where a tempting implication breaks.