🔢Elementary Algebraic Topology Unit 1 – Intro to Topological Spaces

Key Concepts and Definitions

• Topology defined as a collection of open sets on a set $X$ satisfying certain axioms
• Open sets form the basis for defining continuity, connectedness, and compactness in topological spaces
• Closed sets are complements of open sets and play a crucial role in understanding topological properties
• Neighborhoods are open sets containing a given point and help characterize local properties
• Basis is a subcollection of open sets from which all open sets can be generated using unions
  • Subbasis is a collection of open sets from which a basis can be generated using finite intersections
• Hausdorff spaces are topological spaces in which distinct points have disjoint neighborhoods
• Metric spaces are special topological spaces where the topology is induced by a distance function (metric)

Topological Spaces: The Basics

• Topological spaces consist of a set $X$ and a collection of subsets of $X$ called open sets
• Open sets satisfy three axioms:
  1. The empty set $\emptyset$ and the entire set $X$ are open
  2. The union of any collection of open sets is open
  3. The intersection of a finite number of open sets is open
• Closed sets are complements of open sets and satisfy dual axioms
• Closure of a set $A$, denoted $\overline{A}$, is the smallest closed set containing $A$
• Interior of a set $A$, denoted $\text{int}(A)$, is the largest open set contained in $A$
• Boundary of a set $A$, denoted $\partial A$, is the set difference between its closure and interior

Types of Topological Spaces

• Discrete topology on a set $X$ is the collection of all subsets of $X$, making every subset open (and closed)
• Indiscrete topology on a set $X$ is the collection containing only $\emptyset$ and $X$, making only these sets open (and closed)
• Cofinite topology on an infinite set $X$ consists of $\emptyset$ and all subsets of $X$ whose complement is finite
• Usual topology on $\mathbb{R}$ is generated by open intervals $(a,b)$ and forms the basis for metric spaces
• Product topology on a product of topological spaces is generated by products of open sets from each space
• Quotient topology on a quotient space is the largest topology making the quotient map continuous
• CW complex is a topological space constructed by attaching cells of increasing dimension via continuous maps

Continuous Functions and Homeomorphisms

• Continuous function $f: X \to Y$ between topological spaces preserves open sets, i.e., the preimage of an open set is open
• Homeomorphism is a continuous bijection with a continuous inverse, establishing topological equivalence between spaces
• Topological invariants are properties preserved by homeomorphisms (connectedness, compactness, Hausdorff property)
• Topological properties are characteristics of spaces that are invariant under homeomorphisms
• Quotient maps are surjective continuous maps that induce the quotient topology on the codomain
• Homotopy is a continuous deformation of one continuous function into another, capturing the idea of topological similarity
  • Homotopy equivalence is a weaker notion of topological equivalence compared to homeomorphism

Subspaces and Product Spaces

• Subspace topology on a subset $A$ of a topological space $X$ consists of the intersection of open sets in $X$ with $A$
• Product topology on a product of topological spaces $\prod_{i \in I} X_i$ is generated by products of open sets from each $X_i$
• Tychonoff's theorem states that the product of compact spaces is compact in the product topology
• Projection maps from a product space to its factors are continuous and open
• Box topology on a product of topological spaces is generated by products of open sets, but may not be equal to the product topology
• Disjoint union topology on a disjoint union of topological spaces is generated by the union of open sets from each space

Connectedness and Compactness

• Connected space cannot be expressed as the disjoint union of two non-empty open sets
• Path-connected space has a continuous path between any two points
  • Path-connectedness implies connectedness, but the converse is not always true
• Component of a space is a maximal connected subspace
• Compact space has the property that every open cover has a finite subcover
  • Heine-Borel theorem characterizes compact subsets of Euclidean space as closed and bounded
• Locally compact space has the property that every point has a compact neighborhood
• Sequentially compact space has the property that every sequence has a convergent subsequence
  • In metric spaces, sequential compactness is equivalent to compactness

Applications and Examples

• Topological data analysis uses techniques from algebraic topology to study the shape and structure of data sets
• Knot theory studies embeddings of circles in 3-dimensional space up to continuous deformation (ambient isotopy)
  • Knot invariants, such as the fundamental group of the knot complement, help distinguish different knots
• Manifolds are locally Euclidean topological spaces that form the foundation for differential geometry and general relativity
  • Classification of closed surfaces is a fundamental result in 2-dimensional topology
• Algebraic topology uses algebraic structures (groups, rings, modules) to study topological spaces
  • Fundamental group, homology, and cohomology are key invariants in algebraic topology
• Topological quantum field theories (TQFTs) associate algebraic objects to manifolds and study their properties
  • TQFTs have applications in physics, such as the study of quantum gravity and topological phases of matter

Common Pitfalls and Tips

• Carefully verify that a collection of subsets satisfies the axioms for a topology before proceeding with proofs
• Be mindful of the difference between open and closed sets, as well as their complements
• Understand the relationships between different topological properties (Hausdorff, normal, regular, completely regular)
• Use the definitions of continuity, homeomorphism, and homotopy consistently to avoid errors in proofs
• Exploit the interplay between topology and other areas of mathematics (analysis, algebra, geometry) to gain insights
• Develop intuition by studying examples and counterexamples of topological spaces and their properties
  • Visualize spaces using diagrams, simplicial complexes, or CW complexes when possible
• Practice problem-solving techniques, such as using the definitions, constructing examples, and proof by contradiction
• Engage with the material actively by asking questions, discussing with peers, and exploring additional resources