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 ↓


Connectedness


Connectedness is a fundamental concept in topology — a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. In topology, connectedness helps us understand how a space is put together. Specifically, it asks whether a space can be partitioned into two disjoint open sets. This notion captures the intuitive concept of whether a space is "all in one piece."

Basic concepts

To understand the idea of connectivity, we first need to understand some key concepts of topology, such as topological spaces, open sets, and continuous functions.

Topological spaces

A topological space is a set S equipped with a collection of open subsets T that satisfy certain axioms. These axioms include:

  • The empty set and the complete space S are in T.
  • Any union of members of T is also a member of T.
  • The intersection of a finite number of members of T is in T.

This collection T is called the topology on S. The elements of S are called points.

Open and closed sets

Sets in a topology T are called open sets. A set can be open, closed, both, or neither. A set is closed if its complement is open. In the finite topology, the whole set and the empty set are both open and closed, often called clopen sets.

Continuous work

A function between two topological spaces is continuous if the preimage of every open set is open. Continuity in topology generalizes the idea of a continuous function in calculus.

Defining engagement

A topological space is considered connected if it cannot be partitioned into two nonempty disjoint open subsets. In other words, a space is connected if there is no pair of open sets U and V such that:

  • U ∪ V = X (where X is the location),
  • U ∩ V = ∅ (the empty set),
  • Both U and V are not empty.

If such open sets exist, then the space is disjoint.

    Space X is connected ⇔ ¬∃(U, V : U ∪ V = X, U ∩ V = ∅, U ≠ ∅, V ≠ ∅)
    Space X is connected ⇔ ¬∃(U, V : U ∪ V = X, U ∩ V = ∅, U ≠ ∅, V ≠ ∅)

Visual example 1: Straight line interval

Connected intervals

The red dots indicate the endpoints of the interval on the real line. The interval is a connected space because there is no way to partition it into two non-empty open sets that do not overlap.

Lesson example 1: Real number line

Consider the real number line ℝ. This space is connected because you cannot partition it into two non-empty open intervals. Such a continuous category has no intervals or partitions that would divide the space into disjoint subsets.

Visual example 2: Disconnected set

Component A Component B

This illustration shows two separated circles, representing two components of a disjoint space.

Lesson example 2: Disconnected set

Imagine a set consisting of two points on a plane, these are not connected and there exist such open sets that they cover the whole space without intersecting each other, thus they are disjoint.

Component and path association

If a space is not connected, it can be decomposed into a collection of maximal connected subsets, called components. Each component is itself a connected space.

Path affiliation

A space is called path-connected if any two points can be connected by a continuous path. While all path-connected spaces are connected, not all connected spaces are path-connected.

Visual example 3: Path combination

Line-connected path

It shows two circles that are connected by a path. Hence, the whole structure is connected by a path and is also linked.

Text example 3: Non-path connected but connected

Imagine a set that behaves like the topologist's sine curve, which is connected but does not allow paths between the endpoints of the boundaries because of oscillations. This example shows that connected and path-connected are not the same.

Properties of connected spaces

The notion of relatedness has several important properties:

  • Product of connected spaces: The Cartesian product of connected spaces is connected.
  • Images under continuous functions: The image of a connected space under a continuous function remains connected.
  • Components are closed: The components of a space are closed sets.

Visual example 4: Connected products

Connected space A Connected space B

While each rectangle represents a connected space, the Cartesian product preserves connectedness.

Applications and further considerations

Connectedness is central in various areas of mathematics, such as analysis, where understanding continuity and convergence translates into understanding connectedness. Connectedness also plays an important role in complex analysis, algebraic topology, and manifold theory.

In short, the study of connectedness provides important insights into how space, continuity, and boundaries interact. By emphasizing the continuity and seamlessness of mathematical spaces, it lays the groundwork for a more nuanced understanding of spatial relations in higher mathematics.

Conclusion

Connectedness, though seemingly simple, has profound implications in mathematics. By establishing whether a topological space is connected, mathematicians gain insight into the fundamental properties and behaviors of the space. Through the study of connectedness, one also develops a strong understanding of the broader principles that make topology an essential mathematical discipline.


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