# First Extended Atlas Slice: Topology, Number Theory, Decision Problems, and Formal Statements

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



## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%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



## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%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



## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%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



## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%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



## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%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



## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%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 2 · optional



## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections.

[Related questions](/en/explore/?mko=mko-topological-space) [Concept guides](/en/derivations/?q=mko-topological-space)


- Stage 3 · optional



## [Natural-number divisibility](/en/mko/mko-natural-divisibility/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished.

[Related questions](/en/explore/?mko=mko-natural-divisibility) [Concept guides](/en/derivations/?q=mko-natural-divisibility)


- Stage 4 · optional



## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

The natural number gcd(a,b) divides both a and b and is divisible by every common divisor; the inputs are coprime when their gcd is 1.

[Related questions](/en/explore/?mko=mko-gcd-coprime) [Concept guides](/en/derivations/?q=mko-gcd-coprime)


- Stage 5 · optional



## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

A prime natural number is at least 2 and has no positive divisors other than 1 and itself.

[Related questions](/en/explore/?mko=mko-prime-number) [Concept guides](/en/derivations/?q=mko-prime-number)


- Stage 6 · optional



## [Natural-number congruence modulo a positive modulus](/en/mko/mko-modular-congruence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

For a fixed positive modulus n, two natural numbers a and b are congruent modulo n when they have the same remainder on division by n.

[Related questions](/en/explore/?mko=mko-modular-congruence) [Concept guides](/en/derivations/?q=mko-modular-congruence)


- Stage 7 · optional



## [Decision problem](/en/mko/mko-decision-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

A decision problem specifies an input set X and a yes-instance subset Y⊆X; a decider must terminate on every legal input and return the correct Boolean answer.

[Related questions](/en/explore/?mko=mko-decision-problem) [Concept guides](/en/derivations/?q=mko-decision-problem)


- Stage 8 · optional



## [Formal statements in a proof assistant](/en/mko/mko-lean-formal-statement/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261018%2F)

A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context.

[Related questions](/en/explore/?mko=mko-lean-formal-statement) [Concept guides](/en/derivations/?q=mko-lean-formal-statement)



Path source & statuspath-pulse-foundations-20261018 · candidate

[Original paths JSON](/data/architecture/learning-paths.json)