# Wave5 Roots: Subspaces, Topological Covers, Almost-Sure Convergence, Matching, Totient and Finite Automata Complete seven traceable knowledge objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 1 · required ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subset) [Concept guides](/en/derivations/?q=mko-subset) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Open and closed sets](/en/mko/mko-open-closed-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-open-closed-set) [Concept guides](/en/derivations/?q=mko-open-closed-set) - Stage 1 · required ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-space) [Concept guides](/en/derivations/?q=mko-topological-space) - Stage 1 · required ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 1 · required ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 1 · required ## [Sigma-algebras and measure spaces](/en/mko/mko-sigma-algebra-measure-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sigma-algebra-measure-space) [Concept guides](/en/derivations/?q=mko-sigma-algebra-measure-space) - Stage 1 · required ## [Convergence in probability](/en/mko/mko-convergence-in-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-convergence-in-probability) [Concept guides](/en/derivations/?q=mko-convergence-in-probability) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph) - Stage 1 · required ## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-prime-number) [Concept guides](/en/derivations/?q=mko-prime-number) - Stage 1 · required ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-gcd-coprime) [Concept guides](/en/derivations/?q=mko-gcd-coprime) - Stage 1 · required ## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-counting-principle) [Concept guides](/en/derivations/?q=mko-counting-principle) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Abstract computational model](/en/mko/mko-computational-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-computational-model) [Concept guides](/en/derivations/?q=mko-computational-model) - Stage 1 · required ## [Discrete-time dynamical system](/en/mko/mko-discrete-dynamical-system/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-discrete-dynamical-system) [Concept guides](/en/derivations/?q=mko-discrete-dynamical-system) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 2 · optional ## [Real vector subspace](/en/mko/mko-vector-subspace/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar. [Related questions](/en/explore/?mko=mko-vector-subspace) [Concept guides](/en/derivations/?q=mko-vector-subspace) - Stage 3 · optional ## [Topological closure and interior](/en/mko/mko-closure-and-interior/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology. [Related questions](/en/explore/?mko=mko-closure-and-interior) [Concept guides](/en/derivations/?q=mko-closure-and-interior) - Stage 4 · optional ## [Open cover and subcover](/en/mko/mko-open-cover-and-subcover/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target. [Related questions](/en/explore/?mko=mko-open-cover-and-subcover) [Concept guides](/en/derivations/?q=mko-open-cover-and-subcover) - Stage 5 · optional ## [Almost-sure convergence](/en/mko/mko-almost-sure-convergence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero. [Related questions](/en/explore/?mko=mko-almost-sure-convergence) [Concept guides](/en/derivations/?q=mko-almost-sure-convergence) - Stage 6 · optional ## [Graph matching](/en/mko/mko-graph-matching/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed. [Related questions](/en/explore/?mko=mko-graph-matching) [Concept guides](/en/derivations/?q=mko-graph-matching) - Stage 7 · optional ## [Euler totient function](/en/mko/mko-euler-totient-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1. [Related questions](/en/explore/?mko=mko-euler-totient-function) [Concept guides](/en/derivations/?q=mko-euler-totient-function) - Stage 8 · optional ## [Deterministic finite automaton](/en/mko/mko-finite-automaton/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261101%2F) A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states. [Related questions](/en/explore/?mko=mko-finite-automaton) [Concept guides](/en/derivations/?q=mko-finite-automaton) Path source & statuspath-pulse-foundations-20261101 · candidate [Original paths JSON](/data/architecture/learning-paths.json)