Editing Topology (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Topology''' — The mathematical study of properties of spaces preserved under continuous deformations (homeomorphisms).
* '''Topological space''' — A set X with a collection of subsets (the open sets) satisfying: ∅ and X are open; arbitrary unions of open sets are open; finite intersections are open.
* '''Homeomorphism''' — A bijective continuous map with a continuous inverse; the isomorphism of topology; topologically equivalent spaces.
* '''Open set / closed set''' — Fundamental building blocks; open sets are the "neighborhoods" defining the topology.
* '''Basis (topology)''' — A collection of open sets from which all open sets can be generated by unions.
* '''Continuity (topological)''' — f is continuous iff preimages of open sets are open; generalizes ε-δ continuity.
* '''Connectedness''' — A space is connected if it cannot be partitioned into two disjoint non-empty open sets.
* '''Path-connectedness''' — Any two points can be joined by a continuous path; stronger than connectedness.
* '''Compactness''' — Every open cover has a finite subcover; generalizes closed-and-bounded in ℝⁿ.
* '''Homotopy''' — A continuous deformation of one map (or space) into another.
* '''Fundamental group π₁''' — The group of loops (up to homotopy) based at a point; captures 1-dimensional holes.
* '''Homology groups''' — Algebraic invariants measuring n-dimensional holes in a space; H₀, H₁, H₂, ...
* '''Euler characteristic (χ)''' — A topological invariant: χ = V − E + F for polyhedra; χ(sphere) = 2, χ(torus) = 0.
* '''Manifold''' — A topological space that locally looks like Euclidean space; curves are 1-manifolds, surfaces are 2-manifolds.
* '''Genus''' — The number of handles on a surface; torus has genus 1, sphere has genus 0.
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