PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Topology → General Topology ↓


Continuity


Continuity is a fundamental concept in mathematics, especially in analysis and topology, where it describes the behavior of functions. In general topology, continuity extends beyond the familiar setting of the real numbers to more abstract spaces. Understanding continuity in topology requires understanding several related concepts, including open sets, topological spaces, and continuous functions.

Basic concepts

Let's start with the most important things. In topology, we often deal with sets, specifically topological spaces. A topological space is a set equipped with a collection of open subsets that satisfy certain properties:

  • The empty set and the whole space are open.
  • Any union of open sets is open.
  • Any finite intersection of open sets is open.

With these open sets, we can talk about continuity in a more general sense than in calculus. In the world of topology, a function (f: X to Y) between two topological spaces (X) and (Y) is continuous if the preimage of every open set in (Y) is open in (X).

Visual example 1: open set

X You Y V

In the above example, ( U ) is an open set in ( X ), and ( V ) is an open set in ( Y ). For a function ( f: X to Y ) to be continuous, whenever ( V ) is open in ( Y ), then its preimage ( f^{-1}(V) ), which may be ( U ), must be open in ( X ).

The search for continuity

To understand continuity of functions on a topological space more deeply, consider some properties and intuitions:

Continuity and open sets

Consider the function ( f: X to Y ). To ensure the function is continuous, instead of checking all possible points, we focus on open sets in ( Y ) and their pre-images in ( X ). If all these pre-images are open, then the function is continuous.

Example: Suppose ( f: mathbb{R} to mathbb{R} ) is the square function ( f(x) = x^2 ). 
The set ( (1, 4) ) in ( mathbb{R} ) is open. 
The preceding diagram ( f^{-1}((1, 4)) = (-2, -1) cup (1, 2) ) is also open. 
Thus, ( f ) is continuous.

Continuum and closed sets

An equivalent characterization of continuity involves closed sets. A function ( f: X to Y ) is continuous if the preimage of every closed set in ( Y ) is closed in ( X ). This is due to the topological concept that the complement of an open set is closed and vice versa.

Understanding with limitations

Although topology does not require a metric or distance, one can gain intuition by remembering limits from calculus. For a function ( f: mathbb{R} to mathbb{R} ) to be continuous at a point ( x_0 ), the limit of ( f(x) ) as ( x ) gets close to ( x_0 ) must be equal to ( f(x_0) ). In topological terms, as you get arbitrarily close to a point, the images under ( f ) stay close to the image of that point.

Visual example 2: limits and continuity

X f(x)

In the above graph, as ( x ) (red point) gets closer to a certain point, the value of ( f(x) ) (green point) approaches the corresponding point on ( f ). This visualization conveys the essence of continuity: no sudden jumps in function values.

Uniform continuity

Another form of continuity is uniform continuity. A function ( f: X to Y ) is uniformly continuous if for any small allowable change in the range, no matter where you start in the domain, the same small change in the domain works. This is stronger than regular continuity because it does not depend on any particular point.

Lesson example: uniform continuity

Example: Consider ( f(x) = x^2 ) on ( [0, 1] ). 
If given any ( epsilon > 0 ), choose ( delta = sqrt{epsilon + 1} - 1 ). 
Then for all ( x, y in [0, 1] ), 
If ( |x - y| < delta ), then ( |x^2 - y^2| < epsilon ).
Thus, ( f(x) = x^2 ) is uniformly continuous on ( [0, 1] ).

Continuity and homeomorphisms

In topology, not only is continuity important, but so is the idea of homeomorphism. Two spaces are homeomorphic if there exists a continuous bijection between them with a continuous inverse, meaning that the two spaces are essentially the same from a topological perspective.

Visual example 3: homeomorphism

Circle Oval

The circle can be deformed into an ellipse without cutting or pasting. Therefore, the circle and the ellipse are isomorphic. Under the right function, they are consistent in their topological properties, thanks to continuity in both directions.

Topology and real functions

The study of continuity in topology lays the groundwork for understanding more complicated real functions. For example, piecewise functions made up of continuous segments can also be continuous if handled correctly at boundaries.

Example: Consider ( f(x) ) defined as follows:
  ( f(x) = x^2 ) for ( x leq 1 ),
  ( f(x) = 2x - 1 ) where ( x > 1 ).
Check for continuity in the vicinity of ( x = 1 ):
  (lim_{x to 1^-} f(x) = 1), (lim_{x to 1^+} f(x) = 1),
  and ( f(1) = 1 ).

Since the limits on both sides are equal to the value of the function at ( x = 1 ),
( f(x) ) is continuous at ( x = 1 ) and hence integrable.

Concluding remarks

Continuity in topology, while abstract, opens the door to understanding structures beyond simple lines and curves. By considering seamless transformation possibilities between open and closed sets and spaces, topology provides a comprehensive framework for understanding complex mathematical scenarios.

Through extensive examples, visualizations, and explorations into the behavior of functions and spaces, continuity emerges not only as a concept of persistence in practice, but also as a cornerstone of the beautiful symmetries and connections underlying a variety of mathematical disciplines.


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Continuity

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