Mathematics
The Cut Point Trick: A Fast Way to Tell Spaces Apart
A clear guide to cut points, why homeomorphisms preserve them, how the cut point trick works, and why higher-dimensional Euclidean spaces require stronger invariants.
Writing
Blog
8 posts tagged "Topology"
Mathematics
Homomorphic vs Homeomorphic vs Isomorphic
A focused guide to the difference between homomorphic, homeomorphic, and isomorphic, with examples from algebra and topology.
Mathematics
Brouwer's Fixed-Point Theorem: A Proof via the Fundamental Group
We prove Brouwer's Fixed-Point Theorem for the closed disk using the fundamental group and the non-existence of a retraction, deriving an algebraic contradiction.
Mathematics
The Stone-Weierstrass Theorem: Approximating Continuous Functions
We state and prove the Stone-Weierstrass theorem — the powerful generalization of Weierstrass's classical approximation theorem that characterizes when subalgebras of continuous functions are dense — and explore its wide-ranging applications.
Mathematics
Grigori Perelman: The Genius Who Refused a Million Dollars
The remarkable story of Grigori Perelman, the reclusive Russian mathematician who proved the Poincaré conjecture — one of the great unsolved problems in mathematics — and then refused the Fields Medal and the million-dollar Millennium Prize.
Mathematics
The Poincaré Conjecture: From Topology to a Millennium Prize
We explore the Poincaré Conjecture — every simply connected, closed 3-manifold is homeomorphic to the 3-sphere — its proof by Grigori Perelman using Ricci flow, and its place as the first solved Millennium Prize Problem.
Mathematics
Henri Poincaré: The Last Universalist
The life and vast mathematical legacy of Henri Poincaré — the last mathematician who mastered virtually all branches of mathematics and made foundational contributions to topology, dynamical systems, and mathematical physics.
Mathematics
The Heine-Borel Theorem: Why Compactness Matters in Analysis
We prove the Heine-Borel theorem — that a subset of Euclidean space is compact if and only if it is closed and bounded — and explore why compactness is one of the most powerful concepts in all of analysis.