# Second Extended Atlas Slice: Topological Language, Computational Correctness, Asymptotic Bounds, and Proof Terms 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 ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Set membership](/en/mko/mko-set-membership/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-membership) [Concept guides](/en/derivations/?q=mko-set-membership) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 1 · required ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number) - Stage 1 · required ## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-proposition) [Concept guides](/en/derivations/?q=mko-proposition) - Stage 1 · required ## [Mathematical proof](/en/mko/mko-proof/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-proof) [Concept guides](/en/derivations/?q=mko-proof) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%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 ## [Formal statements in a proof assistant](/en/mko/mko-lean-formal-statement/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-lean-formal-statement) [Concept guides](/en/derivations/?q=mko-lean-formal-statement) - Stage 2 · optional ## [Open and closed sets](/en/mko/mko-open-closed-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed. [Related questions](/en/explore/?mko=mko-open-closed-set) [Concept guides](/en/derivations/?q=mko-open-closed-set) - Stage 3 · optional ## [Neighborhood](/en/mko/mko-neighborhood/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open. [Related questions](/en/explore/?mko=mko-neighborhood) [Concept guides](/en/derivations/?q=mko-neighborhood) - Stage 4 · optional ## [Continuous maps between topological spaces](/en/mko/mko-topological-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) A map is topologically continuous when every open set in its target has an open preimage in its source. [Related questions](/en/explore/?mko=mko-topological-continuity) [Concept guides](/en/derivations/?q=mko-topological-continuity) - Stage 5 · optional ## [Abstract computational model](/en/mko/mko-computational-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set. [Related questions](/en/explore/?mko=mko-computational-model) [Concept guides](/en/derivations/?q=mko-computational-model) - Stage 6 · optional ## [Algorithm correctness](/en/mko/mko-algorithm-correctness/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Algorithm correctness separates partial correctness from termination; total correctness requires both. [Related questions](/en/explore/?mko=mko-algorithm-correctness) [Concept guides](/en/derivations/?q=mko-algorithm-correctness) - Stage 7 · optional ## [Asymptotic resource bounds](/en/mko/mko-asymptotic-resource-bound/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail. [Related questions](/en/explore/?mko=mko-asymptotic-resource-bound) [Concept guides](/en/derivations/?q=mko-asymptotic-resource-bound) - Stage 8 · optional ## [Proof term](/en/mko/mko-lean-proof-term/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261019%2F) A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel. [Related questions](/en/explore/?mko=mko-lean-proof-term) [Concept guides](/en/derivations/?q=mko-lean-proof-term) Path source & statuspath-pulse-foundations-20261019 · candidate [Original paths JSON](/data/architecture/learning-paths.json)