🔢Elementary Algebraic Topology Unit 2 – Continuous Functions & Homeomorphisms

Key Concepts

• Continuous functions preserve topological properties such as connectedness and compactness
• Homeomorphisms are bijective continuous functions with continuous inverses
• Topological spaces are considered equivalent if there exists a homeomorphism between them
• Continuity is a fundamental concept in topology, generalizing the notion of continuity from calculus
• Topological invariants are properties preserved under homeomorphisms (Euler characteristic, fundamental group)
• Homotopy is a continuous deformation of one continuous function into another
• Quotient spaces are obtained by identifying certain points or subspaces of a topological space
• Product topology combines two topological spaces into a new space using the Cartesian product

Definitions and Terminology

• Continuous function: A function $f: X \to Y$ between topological spaces is continuous if the preimage of every open set in $Y$ is open in $X$
• Homeomorphism: A bijective continuous function $f: X \to Y$ with a continuous inverse $f^{-1}: Y \to X$
• Topological space: A set $X$ together with a collection of subsets $\tau$ (topology) that satisfies certain axioms (contains empty set and $X$, closed under arbitrary unions and finite intersections)
• Open set: A subset $U$ of a topological space $X$ is open if it belongs to the topology $\tau$
• Closed set: A subset $F$ of a topological space $X$ is closed if its complement $X \setminus F$ is open
• Compact space: A topological space where every open cover has a finite subcover
• Connected space: A topological space that cannot be expressed as the union of two disjoint non-empty open sets
• Path-connected space: A topological space where any two points can be connected by a continuous path

Properties of Continuous Functions

• Continuous functions map connected sets to connected sets
• Composition of continuous functions is continuous ($f \circ g$ is continuous if $f$ and $g$ are continuous)
• Continuous functions preserve compactness (image of a compact set under a continuous function is compact)
• Continuous functions commute with unions and intersections of sets
• If $f: X \to Y$ is continuous and $A \subseteq X$, then $f(\overline{A}) \subseteq \overline{f(A)}$ (closure of the image is contained in the image of the closure)
• Continuous functions preserve convergence of sequences ($x_n \to x$ implies $f(x_n) \to f(x)$)
• Continuous functions on a compact domain attain their maximum and minimum values

Homeomorphisms Explained

• Homeomorphisms are continuous bijections with continuous inverses
• Two spaces are homeomorphic (topologically equivalent) if there exists a homeomorphism between them
• Homeomorphisms preserve all topological properties (connectedness, compactness, separation axioms)
• Composition of homeomorphisms is a homeomorphism
• The inverse of a homeomorphism is a homeomorphism
• Homeomorphic spaces have the same topological invariants (Euler characteristic, fundamental group)
• Examples of homeomorphic spaces:
  • Open interval $(0, 1)$ and the real line $\mathbb{R}$
  • Circle $S^1$ and the annulus $\{(x, y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4\}$
• Spaces that are not homeomorphic:
  • Open interval $(0, 1)$ and closed interval $[0, 1]$
  • Disk $D^2$ and the sphere $S^2$

Examples and Applications

• Möbius strip is a non-orientable surface obtained by identifying the opposite edges of a rectangle with a twist
• Torus (donut shape) is a quotient space obtained from a square by identifying opposite edges
• Stereographic projection is a homeomorphism between the sphere (minus a point) and the Euclidean plane
• Knot theory studies embeddings of circles in 3-dimensional space up to continuous deformation (ambient isotopy)
• Topological data analysis uses persistent homology to study the shape and structure of high-dimensional datasets
• Topological quantum field theories assign algebraic invariants to manifolds and study their properties
• Topological insulators are materials that conduct electricity on their surface but are insulators in their interior due to topological properties of their band structure

Theorems and Proofs

• Intermediate Value Theorem: If $f: [a, b] \to \mathbb{R}$ is continuous and $y$ lies between $f(a)$ and $f(b)$, then there exists $c \in [a, b]$ such that $f(c) = y$
• Brouwer Fixed Point Theorem: Every continuous function from a closed ball to itself has a fixed point
• Tietze Extension Theorem: If $A \subseteq X$ is closed and $f: A \to \mathbb{R}$ is continuous, then there exists a continuous extension $F: X \to \mathbb{R}$ such that $F|_A = f$
• Urysohn's Lemma: In a normal topological space, any two disjoint closed sets can be separated by a continuous function
• Borsuk-Ulam Theorem: For any continuous function $f: S^n \to \mathbb{R}^n$, there exists a point $x \in S^n$ such that $f(x) = f(-x)$
• Jordan Curve Theorem: Every simple closed curve in the plane divides the plane into two regions (interior and exterior)

Problem-Solving Strategies

• To prove a function is continuous, show that the preimage of every open set is open
• To prove two spaces are homeomorphic, construct a bijective continuous function and show its inverse is continuous
• Use topological invariants (Euler characteristic, fundamental group) to distinguish non-homeomorphic spaces
• Employ the lifting criterion to study covering spaces and their properties
• Utilize the compactness and connectedness of spaces to derive properties of continuous functions defined on them
• Apply the theorems (Intermediate Value Theorem, Brouwer Fixed Point Theorem) to solve problems involving continuous functions
• Construct continuous functions using pasting lemmas and partition of unity
• Analyze the local properties of a space (local connectedness, local compactness) to infer global properties

Connections to Other Topics

• Continuity is a fundamental concept in calculus, real analysis, and functional analysis
• Topological spaces generalize metric spaces by focusing on open sets instead of distances
• Algebraic topology uses algebraic structures (groups, rings, modules) to study topological spaces
• Differential topology combines smooth manifolds and continuous maps to study geometric structures
• Topological data analysis applies algebraic topology to study high-dimensional datasets and their shape
• Topological quantum field theories connect topology with quantum physics and category theory
• Knot theory is related to low-dimensional topology and has applications in physics and chemistry
• Dynamical systems and chaos theory study the long-term behavior of continuous functions and their iterates