OCME · Bilingual mathematics & derivations
Second Extended Atlas Slice: Topological Language, Computational Correctness, Asymptotic Bounds, and Proof Terms
On this page
- Topological space
- Subset
- Set operations
- Set membership
- Function mapping
- Set
- Binary relations and set projections
- Natural number
- Proposition
- Mathematical proof
- Arithmetic operations
- Usual order on the real numbers
- Formal statements in a proof assistant
- Open and closed sets
- Neighborhood
- Continuous maps between topological spaces
- Abstract computational model
- Algorithm correctness
- Asymptotic resource bounds
- Proof term
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.
Topological space
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Set operations
Existing canonical prerequisite for the new objects in this pulse.
Set membership
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Set
Existing canonical prerequisite for the new objects in this pulse.
Binary relations and set projections
Existing canonical prerequisite for the new objects in this pulse.
Natural number
Existing canonical prerequisite for the new objects in this pulse.
Proposition
Existing canonical prerequisite for the new objects in this pulse.
Mathematical proof
Existing canonical prerequisite for the new objects in this pulse.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Usual order on the real numbers
Existing canonical prerequisite for the new objects in this pulse.
Formal statements in a proof assistant
Existing canonical prerequisite for the new objects in this pulse.
Open and closed sets
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.
Neighborhood
A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open.
Continuous maps between topological spaces
A map is topologically continuous when every open set in its target has an open preimage in its source.
Abstract computational model
A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set.
Algorithm correctness
Algorithm correctness separates partial correctness from termination; total correctness requires both.
Asymptotic resource bounds
Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail.
Proof term
A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel.
Path source & status
path-pulse-foundations-20261019 · candidate