Competencies: Mathematics > Topology

Topology

Body of Knowledge

Topic	Description	Relevance	Career Tracks
Topological Spaces	Definition via open sets, axioms (arbitrary unions, finite intersections), examples (discrete, indiscrete, standard topology on R^n). Foundation for all topological reasoning.	High	Research Mathematician, Data Scientist, Physicist
Open & Closed Sets	Interior, closure, boundary, limit points. Complements between open and closed. Dense and nowhere-dense sets.	High	Research Mathematician, Data Scientist
Continuity in Topological Setting	Preimage definition of continuity, equivalence with epsilon-delta in metric spaces, continuous images of connected/compact sets.	High	Research Mathematician, ML Engineer
Compactness	Open cover definition, finite subcover, sequential compactness, Heine-Borel theorem, compactness in metric spaces, Tychonoff’s theorem for products.	Critical	Research Mathematician, Physicist, Data Scientist
Connectedness	Connected and path-connected spaces, connected components, intermediate value theorem as consequence, locally connected spaces.	High	Research Mathematician, Data Scientist
Separation Axioms	T0 (Kolmogorov), T1 (Frechet), T2 (Hausdorff), regular, normal spaces. Urysohn’s lemma, Tietze extension theorem.	Medium	Research Mathematician
Product & Quotient Spaces	Product topology (box vs. product), quotient topology and identification maps, examples (torus, Mobius band, projective plane).	High	Research Mathematician, Physicist
Hausdorff Spaces	Uniqueness of limits, closed diagonal characterization, subspaces and products of Hausdorff spaces, compact subsets are closed.	High	Research Mathematician, Physicist
Fundamental Group	Homotopy of paths, loop space, group structure, pi_1(S^1) = Z, van Kampen’s theorem, simply connected spaces.	Medium	Research Mathematician, Physicist
Covering Spaces	Covering maps, lifting properties, deck transformations, universal cover, classification of coverings via subgroups of pi_1.	Medium	Research Mathematician, Physicist
Simplicial Homology	Simplicial complexes, chain groups, boundary operators, homology groups, Betti numbers, Euler characteristic. Bridge to topological data analysis.	High	Data Scientist, Research Mathematician, ML Engineer

Topic

Description

Relevance

Career Tracks

Topological Spaces

Definition via open sets, axioms (arbitrary unions, finite intersections), examples (discrete, indiscrete, standard topology on R^n). Foundation for all topological reasoning.

High

Research Mathematician, Data Scientist, Physicist

Open & Closed Sets

Interior, closure, boundary, limit points. Complements between open and closed. Dense and nowhere-dense sets.

High

Research Mathematician, Data Scientist

Continuity in Topological Setting

Preimage definition of continuity, equivalence with epsilon-delta in metric spaces, continuous images of connected/compact sets.

High

Research Mathematician, ML Engineer

Compactness

Open cover definition, finite subcover, sequential compactness, Heine-Borel theorem, compactness in metric spaces, Tychonoff’s theorem for products.

Critical

Research Mathematician, Physicist, Data Scientist

Connectedness

Connected and path-connected spaces, connected components, intermediate value theorem as consequence, locally connected spaces.

High

Research Mathematician, Data Scientist

Separation Axioms

T0 (Kolmogorov), T1 (Frechet), T2 (Hausdorff), regular, normal spaces. Urysohn’s lemma, Tietze extension theorem.

Medium

Research Mathematician

Product & Quotient Spaces

Product topology (box vs. product), quotient topology and identification maps, examples (torus, Mobius band, projective plane).

High

Research Mathematician, Physicist

Hausdorff Spaces

Uniqueness of limits, closed diagonal characterization, subspaces and products of Hausdorff spaces, compact subsets are closed.

High

Research Mathematician, Physicist

Fundamental Group

Homotopy of paths, loop space, group structure, pi_1(S^1) = Z, van Kampen’s theorem, simply connected spaces.

Medium

Research Mathematician, Physicist

Covering Spaces

Covering maps, lifting properties, deck transformations, universal cover, classification of coverings via subgroups of pi_1.

Medium

Research Mathematician, Physicist

Simplicial Homology

Simplicial complexes, chain groups, boundary operators, homology groups, Betti numbers, Euler characteristic. Bridge to topological data analysis.

High

Data Scientist, Research Mathematician, ML Engineer

Personal Status

To be populated after initial study and self-assessment.