OCME · Bilingual mathematics & derivations
Wave6 Eleventh Layer: Series Tests, Uniform Limits, Product and Quotient Topologies, Paths and Components
On this page
- Infinite series
- Usual order on the real numbers
- Sequence limits and exact tail cutoffs
- Monotone bounded sequence convergence theorem
- Absolute convergence of series
- Absolute value
- Uniform convergence of functions
- Point values and continuity of real functions
- Function mapping
- Supremum and infimum
- Topological basis
- Topological space
- Set operations
- Binary relations and set projections
- Continuous maps between topological spaces
- Function composition
- Quotient set
- Surjectivity and codomain coverage
- Connected topological space
- Real number
- Intermediate value theorem
- Arithmetic operations
- Equivalence class
- Subset
- Topological closure and interior
- Connected component
- Subspace topology
- Comparison tests for series
- Ratio test for series
- A uniform limit of continuous functions is continuous
- Finite product topology
- Quotient topology
- Path-connected space
- Connected component of a topological space
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.
Infinite series
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.
Sequence limits and exact tail cutoffs
Existing canonical prerequisite for the new objects in this pulse.
Monotone bounded sequence convergence theorem
Existing canonical prerequisite for the new objects in this pulse.
Absolute convergence of series
Existing canonical prerequisite for the new objects in this pulse.
Absolute value
Existing canonical prerequisite for the new objects in this pulse.
Uniform convergence of functions
Existing canonical prerequisite for the new objects in this pulse.
Point values and continuity of real functions
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Supremum and infimum
Existing canonical prerequisite for the new objects in this pulse.
Topological basis
Existing canonical prerequisite for the new objects in this pulse.
Topological space
Existing canonical prerequisite for the new objects in this pulse.
Set operations
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.
Continuous maps between topological spaces
Existing canonical prerequisite for the new objects in this pulse.
Function composition
Existing canonical prerequisite for the new objects in this pulse.
Quotient set
Existing canonical prerequisite for the new objects in this pulse.
Surjectivity and codomain coverage
Existing canonical prerequisite for the new objects in this pulse.
Connected topological space
Existing canonical prerequisite for the new objects in this pulse.
Real number
Existing canonical prerequisite for the new objects in this pulse.
Intermediate value theorem
Existing canonical prerequisite for the new objects in this pulse.
Arithmetic operations
Existing canonical prerequisite for the new objects in this pulse.
Equivalence class
Existing canonical prerequisite for the new objects in this pulse.
Subset
Existing canonical prerequisite for the new objects in this pulse.
Topological closure and interior
Existing canonical prerequisite for the new objects in this pulse.
Connected component
Existing canonical prerequisite for the new objects in this pulse.
Subspace topology
Existing canonical prerequisite for the new objects in this pulse.
Comparison tests for series
Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series.
Ratio test for series
For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive.
A uniform limit of continuous functions is continuous
Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain.
Finite product topology
Open rectangles generate the topology of a finite Cartesian product of topological spaces.
Quotient topology
For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open.
Path-connected space
A space is path connected when every pair of its points is joined by a continuous map from the unit interval.
Connected component of a topological space
The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed.
Path source & status
path-pulse-foundations-20261117 · candidate