{"/e/part-1/topology/": {"t": "Topology", "s": "Study of properties preserved under continuous deformations."}, "/e/part-1/surface-topology/": {"t": "Surface (topology)", "s": "A surface is a two-dimensional topological manifold."}, "/e/part-1/manifold/": {"t": "Manifold", "s": "A topological space locally resembling Euclidean space near each point."}, "https://codexery.com/sets-and-logic/e/part-1/set-mathematics/": {"t": "Set (mathematics)", "s": "A set is a collection of different things, typically mathematical objects."}, "https://codexery.com/set-theory-and-foundations/e/part-1/axiom/": {"t": "Axiom", "s": "A foundational statement accepted as true without proof."}, "https://codexery.com/geometry-and-shapes/e/part-1/space-mathematics/": {"t": "space-mathematics", "s": ""}, "https://codexery.com/geometry-and-topology/e/part-1/connected-space/": {"t": "connected-space", "s": ""}, "https://codexery.com/scientists-and-mathematicians/e/part-1/leonhard-euler/": {"t": "Leonhard Euler", "s": "Swiss polymath who founded graph theory and topology, introduced key mathematical notation, and was the most prolific mathematician of the 18th century."}, "https://codexery.com/historical-figures/e/notable/augustin-louis-cauchy/": {"t": "Augustin-Louis Cauchy", "s": "Pioneer of real and complex analysis, prolific mathematician and engineer."}, "https://codexery.com/historical-figures/e/notable/carl-friedrich-gauss/": {"t": "Carl Friedrich Gauss", "s": "German mathematician, astronomer, geodesist, and physicist"}, "https://codexery.com/mathematicians/e/part-1/felix-klein/": {"t": "Felix Klein", "s": "German mathematician known for the Erlangen program and the Klein bottle.", "i": "https://codexery.com/mathematicians/img/part-1/felix-klein.webp", "c": "via Wikipedia: Felix Klein · see source"}, "https://codexery.com/mathematicians/e/part-1/henri-poincar/": {"t": "Henri Poincaré", "s": "French polymath and last universalist mathematician."}, "https://codexery.com/mathematicians/e/part-1/felix-hausdorff/": {"t": "Felix Hausdorff", "s": "Founder of modern topology and pioneer of set theory", "i": "https://codexery.com/mathematicians/img/part-1/felix-hausdorff.webp", "c": "via Wikipedia: Felix Hausdorff · see source"}, "https://codexery.com/set-theory-and-foundations/e/part-1/axiomatic-system/": {"t": "Axiomatic system", "s": "Axiomatic systems: the deductive backbone of mathematics and logic."}, "https://codexery.com/sets-and-logic/e/part-1/subset/": {"t": "Subset", "s": "A set whose elements all belong to another set."}, "https://codexery.com/set-theory-and-foundations/e/part-1/complement-set-theory/": {"t": "Complement (set theory)", "s": "Set of elements not in a given set."}, "https://codexery.com/set-theory-logic/e/part-1/power-set/": {"t": "Power set", "s": "The set of all subsets of a given set."}, "https://codexery.com/combinatorics/e/part-1/bijection/": {"t": "bijection", "s": ""}, "/e/part-1/homotopy/": {"t": "Homotopy", "s": "A continuous deformation between two functions in topology."}, "https://codexery.com/geometry-and-topology/e/part-1/homology-mathematics/": {"t": "homology-mathematics", "s": ""}, "https://codexery.com/sets-and-logic/e/part-1/ordinal-number/": {"t": "Ordinal number", "s": "Ordinals extend counting to infinite well-ordered sets."}, "/e/part-1/topological-space/": {"t": "Topological space", "s": "A set with structure defining closeness without distance."}, "https://codexery.com/languages-and-dialects/e/part-1/greek-language/": {"t": "Greek language", "s": "Greek is the world's oldest recorded living language."}, "/e/part-1/compact-space/": {"t": "Compact space", "s": "A topological property making spaces behave like finite sets."}, "https://codexery.com/historical-figures/e/notable/gottfried-wilhelm-leibniz/": {"t": "Gottfried Wilhelm Leibniz", "s": "German polymath and co-creator of calculus"}, "/e/part-1/euler-characteristic/": {"t": "Euler characteristic", "s": "Topological invariant describing shape regardless of bending."}, "https://codexery.com/geometry-and-topology/e/part-1/torus/": {"t": "torus", "s": ""}, "https://codexery.com/geometry-and-shapes/e/part-1/polyhedron/": {"t": "polyhedron", "s": ""}, "https://codexery.com/media-and-publishing/e/part-1/nature-journal/": {"t": "Nature (journal)", "s": "British weekly journal of peer-reviewed natural science research.", "i": "https://codexery.com/media-and-publishing/img/part-1/nature-journal.webp", "c": "Salwa Farwaneh · CC BY-SA 4.0"}, "https://codexery.com/geometry-and-topology/e/part-1/algebraic-topology/": {"t": "algebraic-topology", "s": ""}, "https://codexery.com/historical-figures/e/notable/georg-cantor/": {"t": "Georg Cantor", "s": "Founder of set theory and transfinite numbers"}, "/e/part-1/klein-bottle/": {"t": "Klein bottle", "s": "A one-sided surface with no boundary."}, "https://codexery.com/geometry-and-topology/e/part-1/cohomology/": {"t": "cohomology", "s": ""}, "https://codexery.com/geometry-and-topology/e/part-1/differential-geometry/": {"t": "differential-geometry", "s": ""}, "/e/part-1/sheaf-mathematics/": {"t": "Sheaf (mathematics)", "s": "Tool for tracking local data on topological spaces."}, "https://codexery.com/geometry-and-topology/e/part-1/algebraic-geometry/": {"t": "algebraic-geometry", "s": ""}, "https://codexery.com/sets-and-logic/e/part-1/empty-set/": {"t": "Empty set", "s": "The unique set containing no elements."}, "https://codexery.com/languages-and-dialects/e/part-1/french-language/": {"t": "French language", "s": "A Romance language of the Indo-European family, descended from Vulgar Latin and shaped by Celtic and Frankish influences."}, "/e/part-1/fundamental-group/": {"t": "Fundamental group", "s": "A homotopy invariant group of loops in a topological space."}, "/e/part-1/m-bius-strip/": {"t": "Möbius strip", "s": "A non-orientable surface with one side and one boundary."}, "https://codexery.com/geometry-and-shapes/e/part-1/three-dimensional-space/": {"t": "three-dimensional-space", "s": ""}, "https://codexery.com/geometry-and-shapes/e/part-1/solid-geometry/": {"t": "solid-geometry", "s": ""}, "https://codexery.com/geometry-and-shapes/e/part-1/surface-mathematics/": {"t": "surface-mathematics", "s": ""}, "https://codexery.com/geometry-and-topology/e/part-1/coordinate-system/": {"t": "coordinate-system", "s": ""}, "https://codexery.com/lines-hemispheres/e/part-1/latitude/": {"t": "Latitude", "s": "Latitude specifies north-south position on Earth or celestial bodies.", "i": "https://codexery.com/lines-hemispheres/img/part-1/latitude.webp", "c": "source: Pearson Scott Foresman, donated to the Wikimedia Foundation\nsvg conversion: Gregors\nColor: Jahobr · via Wikipedia: Latitude · Public domain"}, "https://codexery.com/lines-hemispheres/e/part-1/longitude/": {"t": "Longitude", "s": "The east-west coordinate that, with latitude, pinpoints any location on Earth.", "i": "https://codexery.com/lines-hemispheres/img/part-1/longitude.webp", "c": "Original:  Pearson Scott Forman Vector:  Gregors · via Wikipedia: Longitude · CC BY-SA 3.0"}, "https://codexery.com/lines-hemispheres/e/part-1/180th-meridian/": {"t": "180th meridian", "s": "The 180th meridian divides hemispheres and anchors the International Date Line."}, "https://codexery.com/design-digital-arts/e/part-1/computer-graphics/": {"t": "computer-graphics", "s": ""}, "https://codexery.com/transportation-engineering/e/part-1/airplane/": {"t": "Airplane", "s": "Fixed-wing aircraft propelled by thrust, from the Wright brothers to modern jets.", "i": "https://codexery.com/transportation-engineering/img/part-1/airplane.webp", "c": "SamKramer123 · CC BY 4.0"}, "https://codexery.com/polish-cuisine/e/part-1/bagel/": {"t": "Bagel", "s": "Boiled-then-baked ring bread from Polish Jewish tradition."}, "https://codexery.com/probability-and-stochastic-processes/e/part-1/variance/": {"t": "Variance", "s": "Variance measures how far numbers spread from their average."}, "https://codexery.com/particle-and-nuclear-physics/e/part-1/string-theory/": {"t": "String theory", "s": "Theoretical framework replacing point particles with vibrating strings."}, "/e/part-1/covering-space/": {"t": "Covering space", "s": "A space that locally projects multiple copies onto a base space."}, "https://codexery.com/combinatorics/e/part-1/combinatorics/": {"t": "combinatorics", "s": ""}, "https://codexery.com/geometry-and-shapes/e/part-1/polygon/": {"t": "polygon", "s": ""}, "https://codexery.com/geometry-and-shapes/e/part-1/star-polygon/": {"t": "star-polygon", "s": ""}, "https://codexery.com/probability-statistics/e/part-1/parameter/": {"t": "Parameter", "s": "A characteristic that helps define or classify a system."}, "https://codexery.com/geometry-and-shapes/e/part-1/unit-circle/": {"t": "unit-circle", "s": ""}, "https://codexery.com/classical-mechanics-and-dynamics/e/part-1/rotation/": {"t": "rotation", "s": ""}, "https://codexery.com/geometry-and-topology/e/part-1/knot-theory/": {"t": "knot-theory", "s": ""}, "https://codexery.com/geometry-and-topology/e/part-1/cartesian-product/": {"t": "cartesian-product", "s": ""}, "https://codexery.com/geometry-and-topology/e/part-1/convex-set/": {"t": "convex-set", "s": ""}, "https://codexery.com/physical-ocean-floor-features/e/part-1/equator/": {"t": "Equator", "s": "Imaginary line dividing Earth into Northern and Southern Hemispheres."}, "https://codexery.com/algebra/e/part-1/normal-subgroup/": {"t": "normal-subgroup", "s": ""}, "https://codexery.com/sets-and-logic/e/part-1/union-set-theory/": {"t": "Union (set theory)", "s": "Set operation combining all elements from given sets."}, "https://codexery.com/mechanics-and-fluid-dynamics/e/part-1/hamiltonian-mechanics/": {"t": "Hamiltonian mechanics", "s": "Reformulation of Lagrangian mechanics using generalized momenta."}, "https://codexery.com/classical-mechanics/e/part-1/classical-mechanics/": {"t": "classical-mechanics", "s": ""}, "https://codexery.com/relativity-and-spacetime/e/part-1/spacetime/": {"t": "Spacetime", "s": "A four-dimensional continuum fusing space and time."}, "https://codexery.com/geometry-and-shapes/e/part-1/square/": {"t": "square", "s": ""}, "https://codexery.com/historical-figures/e/notable/euclid/": {"t": "Euclid", "s": "Ancient Greek mathematician, author of the Elements."}, "https://codexery.com/set-theory-and-foundations/e/part-1/cardinality/": {"t": "Cardinality", "s": "Fundamental concept defining the size of sets."}, "https://codexery.com/transportation/e/part-1/navigation/": {"t": "Navigation", "s": "The art and science of directing movement from one place to another."}, "https://codexery.com/greek-inventions/e/part-1/gimbal/": {"t": "gimbal", "s": ""}, "https://codexery.com/computing-digital/e/part-1/computer-science/": {"t": "computer-science", "s": ""}, "https://codexery.com/historical-figures/e/notable/gustav-kirchhoff/": {"t": "Gustav Kirchhoff", "s": "German physicist and mathematician known for Kirchhoff's circuit laws, spectroscopy, and black-body radiation."}, "https://codexery.com/writing-systems/e/part-1/greek-alphabet/": {"t": "Greek alphabet", "s": "Earliest alphabet to systematically write vowels and consonants.", "i": "https://codexery.com/writing-systems/img/part-1/greek-alphabet.webp", "c": "Unknown author · Public domain"}, "https://codexery.com/geometry-and-shapes/e/part-1/sphere/": {"t": "sphere", "s": ""}, "https://codexery.com/philosophers/e/part-1/imre-lakatos/": {"t": "Imre Lakatos", "s": "Hungarian philosopher of mathematics and science."}, "/e/part-1/betti-number/": {"t": "Betti number", "s": "Topological invariants measuring holes in spaces."}, "https://codexery.com/miscellaneous/e/part-1/belt-mechanical/": {"t": "Belt (mechanical)", "s": "A loop of flexible material linking rotating shafts for power transmission."}, "https://codexery.com/map-projections-cartography-foundations/e/part-1/world-map/": {"t": "World map", "s": "A map of most or all of Earth's surface."}, "https://codexery.com/historical-figures/e/artists/m-c-escher/": {"t": "M. C. Escher", "s": "Dutch graphic artist known for mathematically inspired woodcuts, lithographs, and mezzotints.", "i": "https://codexery.com/historical-figures/img/artists/m-c-escher.webp", "c": "Hans Peters for Anefo · via Wikipedia: M. C. Escher · CC0"}, "https://codexery.com/composers/e/part-1/johann-sebastian-bach/": {"t": "johann-sebastian-bach", "s": ""}, "https://codexery.com/narrative-forms-and-devices/e/part-1/speculative-fiction/": {"t": "Speculative fiction", "s": "Umbrella genre for fiction beyond everyday reality."}, "https://codexery.com/chinese-inventions/e/part-1/chain-pump/": {"t": "chain-pump", "s": ""}, "https://codexery.com/physical-ocean-floor-features/e/part-1/great-circle/": {"t": "Great circle", "s": "Largest circle on a sphere, shortest path between two points."}, "https://codexery.com/geometry-and-topology/e/part-1/affine-transformation/": {"t": "affine-transformation", "s": ""}, "https://codexery.com/chemical-bonding-and-structure/e/part-1/molecule/": {"t": "molecule", "s": ""}, "https://codexery.com/manufacturing-and-materials/e/part-1/recycling/": {"t": "Recycling", "s": "Converting waste into resources for a sustainable future."}, "https://codexery.com/performance-arts/e/part-1/magic-illusion/": {"t": "Magic (illusion)", "s": "Performing art of seemingly impossible feats using natural means."}, "https://codexery.com/geometry-and-shapes/e/part-1/two-dimensional-space/": {"t": "two-dimensional-space", "s": ""}, "https://codexery.com/algorithms-and-data-structures/e/part-1/bisection-method/": {"t": "bisection-method", "s": ""}, "https://codexery.com/historical-figures/e/notable/david-hilbert/": {"t": "David Hilbert", "s": "German mathematician and philosopher of mathematics"}}