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


General topology, also called point-set topology, is a branch of mathematics that studies more abstract aspects of geometric objects. It aims to provide a foundational understanding of continuity, compactness, and convergence using minimal assumptions. It explores how shapes and spaces can be folded or stretched while preserving certain properties.

Basics of general topology

Before delving into deeper topics, we first need to understand some basic concepts such as sets, functions, and relations. A set is a collection of distinct objects, considered as an object in its own right. Sets can be made of anything: numbers, letters, shapes, or even other sets.

Example: The set of integers between 1 to 5 can be written as {1, 2, 3, 4, 5}.

A function is a relation between a set of inputs and a set of acceptable outputs. It is often represented as f: X → Y, where X is the domain and Y is the codomain.

Example: f(x) = x 2 is a function that maps a number to its square, with both domain and codomain being real numbers.

Topological spaces

At the core of general topology lies the concept of a topological space. A topological space is a set equipped with a topology. A topology on a set X is a collection of open subsets of X that includes the empty set and X itself, and is closed under arbitrary unions and finite intersections.

Definition: A topological space is a pair (X, τ), where X is a set and τ is a topology on X.
Example: Consider X = {a, b, c}. The topology on X can be τ = {∅, {a}, {a, b}, {a, b, c}}.

Open and closed sets

In topology, an important distinction is made between open and closed sets. An open set is an element of the topology. A closed set is a set whose complement is open.

Theorem: The intersection of any collection of closed sets is closed, and the union of a finite number of closed sets is closed.

This theorem is a direct consequence of the definition of topology. These properties mimic the familiar properties of open and closed intervals in real analysis.

Continuity and homeomorphisms

A function between two topological spaces is called continuous if the preimage of every open set is open. This generalizes the usual notion of continuity in calculus. Two spaces are homeomorphic if there exists a continuous bijective function with a continuous inverse between them.

Definition: A function f : (X, τ) → (Y, σ) is continuous if for every open set V in σ, the preimage f − 1 (V) is in τ.
Example: The function f: ℝ → ℝ defined by f(x) = 2x is continuous.

Two spaces are considered topologically equivalent or homeomorphic if they can be deformed into each other without tearing or gluing. This idea reflects the essence of "shape" in topology.

Convergence, filters, and nets

Unlike sets of numbers in calculus, sets in topology may not be linearly ordered. Therefore, the concept of a sequence is not always sufficient to capture convergence. Instead, topologists use filters and nets.

Filter

A filter on a set X is a nonempty family F of subsets of X that is closed under finite intersection and superset, and does not contain the empty set.

Net

A net is a generalization of a sequence, but it is indexed by a directed set instead of natural numbers. Each element of the net is mapped to an element of a topological space.

Example: Consider a net directed by a set of integers whose order is normal. f(n)

Compactness

Compactness is a property that generalizes the idea of a subset being closed and bounded in a Euclidean space. Formally, a space is compact if every open cover has a finite subcover.

Theorem (Heine–Borel): A subset of ℝ n is compact if and only if it is closed and bounded.

Compactness plays an important role in analyzing the behavior of functions in topology because it provides a robust form of convergence.

Connectedness

A topological space is connected if it cannot be partitioned into two disjoint non-empty open sets. Intuitively, a space is connected if it is all in one piece.

Example: In the real numbers the interval [0, 1] is connected, but the union of [0, 0.5) and (0.5, 1] is not connected.

Metric space

Metric spaces are special types of topological spaces where a distance function (or metric) is defined. This allows us to introduce the concepts of limits and continuity similar to real analysis.

Definition: A metric space (X, d) is a set X equipped with a function d : X × X → [0, ∞) that satisfies the following:
  • For all x, y in X, d(x, y) = 0 if and only if x = y (the identity of indistinguishable).
  • For all x, y in X, d(x, y) = d(y, x) (isomorphism).
  • For all x, y, z in X, d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
Example: The usual distance function on ℝ is a metric: d(x, y) = |x - y|.

Importance of general topology

Having a solid understanding of general topology is essential for many areas of mathematics, including fields like analysis, geometry, and even data science, where concepts like proximity and continuity appear regularly. It is also important for physics and engineering, in order to model the spaces that appear naturally in these fields.

General topology provides a language and framework for expressing and handling these concepts compactly, and provides tools for understanding complex structures and continuous changes.

Conclusion

General topology emerges as a cornerstone of pure mathematics, laying the groundwork for more specialized fields such as algebraic topology and differential topology. By understanding its fundamentals, mathematicians equip themselves with the tools necessary to explore and understand the infinite fabric of space that characterizes advanced mathematical theory. Whether dealing with the convergence of sequences, the continuity of functions, or the size and boundaries of spaces, general topology is indispensable.


PHD → 3.1


U
username
0%
completed in PHD

← Prev (3)
Topology
Next (3.1.1) →
Open and Closed Sets

Comments 



General Topology

https://www.buddymath.com/phd/3.1?bmal=en