# Wave6 Eleventh Layer: Series Tests, Uniform Limits, Product and Quotient Topologies, Paths and Components 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 ## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-infinite-series) [Concept guides](/en/derivations/?q=mko-infinite-series) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence-limit) [Concept guides](/en/derivations/?q=mko-sequence-limit) - Stage 1 · required ## [Monotone bounded sequence convergence theorem](/en/mko/mko-monotone-sequence-convergence-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-monotone-sequence-convergence-theorem) [Concept guides](/en/derivations/?q=mko-monotone-sequence-convergence-theorem) - Stage 1 · required ## [Absolute convergence of series](/en/mko/mko-absolute-convergence-of-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-convergence-of-series) [Concept guides](/en/derivations/?q=mko-absolute-convergence-of-series) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Uniform convergence of functions](/en/mko/mko-uniform-convergence-of-functions/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-uniform-convergence-of-functions) [Concept guides](/en/derivations/?q=mko-uniform-convergence-of-functions) - Stage 1 · required ## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-continuity) [Concept guides](/en/derivations/?q=mko-continuity) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%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 ## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-supremum-and-infimum) [Concept guides](/en/derivations/?q=mko-supremum-and-infimum) - Stage 1 · required ## [Topological basis](/en/mko/mko-topological-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-basis) [Concept guides](/en/derivations/?q=mko-topological-basis) - Stage 1 · required ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%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-20261117%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 ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-relation) [Concept guides](/en/derivations/?q=mko-relation) - Stage 1 · required ## [Continuous maps between topological spaces](/en/mko/mko-topological-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-topological-continuity) [Concept guides](/en/derivations/?q=mko-topological-continuity) - Stage 1 · required ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 1 · required ## [Quotient set](/en/mko/mko-quotient-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quotient-set) [Concept guides](/en/derivations/?q=mko-quotient-set) - Stage 1 · required ## [Surjectivity and codomain coverage](/en/mko/mko-surjective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-surjective) [Concept guides](/en/derivations/?q=mko-surjective) - Stage 1 · required ## [Connected topological space](/en/mko/mko-connected-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-connected-topological-space) [Concept guides](/en/derivations/?q=mko-connected-topological-space) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Intermediate value theorem](/en/mko/mko-intermediate-value-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-intermediate-value-theorem) [Concept guides](/en/derivations/?q=mko-intermediate-value-theorem) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Equivalence class](/en/mko/mko-equivalence-class/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equivalence-class) [Concept guides](/en/derivations/?q=mko-equivalence-class) - Stage 1 · required ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%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 ## [Topological closure and interior](/en/mko/mko-closure-and-interior/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-closure-and-interior) [Concept guides](/en/derivations/?q=mko-closure-and-interior) - Stage 1 · required ## [Connected component](/en/mko/mko-connected-component/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-connected-component) [Concept guides](/en/derivations/?q=mko-connected-component) - Stage 1 · required ## [Subspace topology](/en/mko/mko-subspace-topology/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-subspace-topology) [Concept guides](/en/derivations/?q=mko-subspace-topology) - Stage 2 · optional ## [Comparison tests for series](/en/mko/mko-series-comparison-test/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series. [Related questions](/en/explore/?mko=mko-series-comparison-test) [Concept guides](/en/derivations/?q=mko-series-comparison-test) - Stage 3 · optional ## [Ratio test for series](/en/mko/mko-ratio-test-series/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive. [Related questions](/en/explore/?mko=mko-ratio-test-series) [Concept guides](/en/derivations/?q=mko-ratio-test-series) - Stage 4 · optional ## [A uniform limit of continuous functions is continuous](/en/mko/mko-uniform-limit-continuity-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain. [Related questions](/en/explore/?mko=mko-uniform-limit-continuity-theorem) [Concept guides](/en/derivations/?q=mko-uniform-limit-continuity-theorem) - Stage 5 · optional ## [Finite product topology](/en/mko/mko-finite-product-topology/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) Open rectangles generate the topology of a finite Cartesian product of topological spaces. [Related questions](/en/explore/?mko=mko-finite-product-topology) [Concept guides](/en/derivations/?q=mko-finite-product-topology) - Stage 6 · optional ## [Quotient topology](/en/mko/mko-quotient-topology/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open. [Related questions](/en/explore/?mko=mko-quotient-topology) [Concept guides](/en/derivations/?q=mko-quotient-topology) - Stage 7 · optional ## [Path-connected space](/en/mko/mko-path-connected-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) A space is path connected when every pair of its points is joined by a continuous map from the unit interval. [Related questions](/en/explore/?mko=mko-path-connected-space) [Concept guides](/en/derivations/?q=mko-path-connected-space) - Stage 8 · optional ## [Connected component of a topological space](/en/mko/mko-topological-connected-component/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261117%2F) The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed. [Related questions](/en/explore/?mko=mko-topological-connected-component) [Concept guides](/en/derivations/?q=mko-topological-connected-component) Path source & statuspath-pulse-foundations-20261117 · candidate [Original paths JSON](/data/architecture/learning-paths.json)