Properties preserved under continuous deformations
← Back to MathematicsLearn the fundamental concepts of topology and topological spaces.
Study continuous functions and topological equivalence between spaces.
Explore the relationship between topology and metric spaces.
Study two fundamental topological properties and their applications.
Learn the hierarchy of separation properties in topological spaces.
Study methods for constructing new topological spaces from existing ones.
Introduction to algebraic topology through the fundamental group.
Study 2-dimensional manifolds and their topological classification.
Explore higher-dimensional topological invariants.
Explore advanced topology and its applications to other areas.
Build the foundation with the basic concepts and definitions of topology.
Learn that a topology on a set X is a collection τ of subsets satisfying: ∅, X ∈ τ; arbitrary unions in τ; finite intersections in τ.
Understand open sets as elements of the topology and closed sets as complements of open sets. Learn De Morgan's laws in topological context.
Define neighborhoods and neighborhood systems. Learn that open sets are neighborhoods of all their points, and understand neighborhood characterization of topology.
Study int(A) as largest open set in A, cl(A) as smallest closed set containing A, and ∂A = cl(A) ∩ cl(X\A). Learn their properties and relationships.
Learn basis B as collection whose unions generate topology. Study subbasis S where finite intersections of S elements form a basis. Understand topology generation.
Define relative topology on subsets Y ⊆ X as τY = {U ∩ Y : U ∈ τ}. Learn inheritance of topological properties to subspaces.
Study standard topology on ℝ, usual topology on ℝⁿ, finite complement topology, cofinite topology, and topologies on finite sets.
Learn discrete topology (all subsets open) and indiscrete topology (only ∅ and X open). Understand these as extreme cases and their properties.
Study continuous functions and when two topological spaces are essentially the same.
Define f: X → Y continuous if f⁻¹(U) is open in X for every open U in Y. Learn this generalizes ε-δ continuity from analysis.
Learn continuity equivalences: open set definition, closed set characterization, neighborhood preservation, and closure/interior characterizations.
Study homeomorphisms as continuous bijections with continuous inverse. Learn that homeomorphic spaces are topologically equivalent and indistinguishable.
Understand properties preserved by homeomorphisms: compactness, connectedness, separation axioms. Learn to prove spaces are not homeomorphic.
Study maps taking open sets to open sets (open maps) and closed sets to closed sets (closed maps). Learn their relationship to quotient maps.
Learn topological embeddings as homeomorphisms onto their image. Study quotient maps as surjections with quotient topology on codomain.
Study homeomorphisms: open interval ≅ ℝ, stereographic projection S² \ {pt} ≅ ℝ², and Möbius strip constructions.
Understand topology's goal of classifying spaces up to homeomorphism. Learn invariants help distinguish non-homeomorphic spaces.
Explore the relationship between topology and metrics, and when topologies arise from metrics.
Learn metric d: X × X → ℝ satisfying positivity, symmetry, and triangle inequality. Study how metrics induce topologies via open balls.
Define open balls B(x,r) = {y : d(x,y) < r} and metric topology as unions of open balls. Learn relationship to ε-δ definitions.
Study sequence convergence xₙ → x iff d(xₙ,x) → 0. Learn how this relates to topological convergence via neighborhoods.
Learn complete metric spaces where Cauchy sequences converge. Study Baire category theorem: complete metric spaces are Baire spaces.
Study topological spaces whose topology comes from some metric. Learn that metrizable spaces have nice properties like first countability.
Learn Urysohn's theorem: regular spaces with countable basis are metrizable. Understand conditions for metrizability.
Study spaces that cannot arise from metrics: cofinite topology on infinite sets, product of uncountably many copies of [0,1].
Learn uniform spaces as generalization of metric spaces. Study uniform topology and relationship to metrization theory.
Study two fundamental properties that distinguish different types of topological spaces.
Learn X is connected if it cannot be written as union of two non-empty disjoint open sets. Understand connectedness as topological "wholeness."
Study path-connected spaces where any two points can be joined by a continuous path. Learn that path-connected implies connected.
Learn connected components as maximal connected subsets. Study path components and their relationship to connected components.
Prove intermediate value theorem using connectedness: continuous image of connected set is connected. Understand topological generalization.
Define compactness via open covers: every open cover has finite subcover. Learn compactness as topological finiteness condition.
Study characterizations of compactness: finite subcover property, finite intersection property for closed sets, and ultrafilter convergence.
Learn Heine-Borel theorem: subsets of ℝⁿ are compact iff closed and bounded. Study sequential compactness equivalence in metric spaces.
Learn arbitrary products of compact spaces are compact. Understand this fundamental theorem and its equivalence to axiom of choice.
Learn the hierarchy of separation properties that distinguish points and sets in topological spaces.