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


Topology is a fascinating field of mathematics that explores the properties that are preserved through deformation, rotation, and stretching of objects. It is often colloquially referred to as "rubber-sheet geometry," which emphasizes that objects can be stretched and bent without tearing or sticking.

Basic concepts

At the core of topology is the concept of a topological space. A topological space is a set of points, with a set of neighborhoods for each point, satisfying a set of axioms relating points and neighborhoods. This can be informally thought of as a way to talk about continuity and extent without the flavor of distance and measure found in metric spaces.

The basic building blocks of topology include the following concepts:

  • Open and closed sets: Topology is primarily concerned with open and closed sets rather than points. An open set can be intuitively thought of as any subset of a space that does not include its boundary, while a closed set includes the boundary.
  • Continuity: a function is continuous if it maps open sets to open sets. In the context of metric spaces, this definition is similar to but more generalized than the well-known epsilon–delta definition.
  • Homeomorphism: This is an important concept where two spaces are considered equivalent (homeomorphic) if one can be transformed into the other in such a way that it is continuous with a continuous inverse.

A topological property is a property that is preserved under homeomorphisms, such as connectedness or compactness.

Visual example

Let's imagine a simple visual example to understand how topology examines locations.

Consider a coffee cup and a donut. Topologically, these two are considered equivalent because you can transform the coffee cup into a donut through continuous deformation, without tearing or sticking.

Here is a symbolic illustration:

The circle represents the donut, and the path symbolizes the handle of the coffee cup.

Important topological properties

Connectedness

A space is said to be connected if it cannot be partitioned into two disjoint non-empty open sets. Connectedness is about the idea of something being in "one piece". For example, a circle is connected because it is a continuous loop without any breaks.

Example: The set (0, 1) in the real numbers is connected, whereas the set {0, 1} is not.
Example: The set (0, 1) in the real numbers is connected, whereas the set {0, 1} is not.

Density

A space is compact if every open cover has a finite subcover. Essentially, this property generalizes the notion of a set being closed and bounded. The Heine–Borel theorem presents a well-known result regarding compactness in Euclidean space.

Example: The closed interval [0, 1] is compact in the real numbers, while (0, 1) is not.
Example: The closed interval [0, 1] is compact in the real numbers, while (0, 1) is not.

Topological spaces

Let's take a deeper look at what a topological space is. A topological space is a set ( X ) containing a collection ( mathcal{T} ) of subsets of ( X ). The collection ( mathcal{T} ) must satisfy three conditions:

  1. The empty set and ( X ) itself belong to ( mathcal{T} ).
  2. The union of any number of sets in ( mathcal{T} ) is also in ( mathcal{T} ).
  3. The intersection of any finite number of sets in ( mathcal{T} ) also occurs in ( mathcal{T} ).

( (X, mathcal{T}) ) is then referred to as a topological space, and the elements of ( mathcal{T} ) are the open sets of the space.

Examples in various locations

Euclidean space example

Consider the real numbers ( mathbb{R} ) with the standard topology. In this case, the open sets are the open intervals such as ( (a, b) ).

If ( a < b ), then (a, b) forms an open set in the topology of ( mathbb{R} ).
If ( a < b ), then (a, b) forms an open set in the topology of ( mathbb{R} ).

Discrete and univariate topology

Two other simple, yet fundamental topologies on any set (X) are the discrete and inseparable topologies.

Discrete topology

In the discrete topology, every subset of ( X ) is open. This means that the topology is made up of all possible subsets of ( X ).

If ( X = {a, b} ), then the discrete topology is ({emptyset, {a}, {b}, {a, b}}).
If ( X = {a, b} ), then the discrete topology is ({emptyset, {a}, {b}, {a, b}}).

Inseparable topology

The discontinuous topology, sometimes called the trivial topology, has only two open sets: the whole set and the empty set.

If ( X = {a, b} ), then the indiscrete topology is ({emptyset, {a, b}}).
If ( X = {a, b} ), then the indiscrete topology is ({emptyset, {a, b}}).

Convergence and continuity

In topology, unlike metric spaces, convergence and continuity are defined without a notion of distance. A sequence is said to converge to a limit if, for any neighborhood of that limit, there is a point in the sequence where all subsequent points lie within the neighborhood.

Continuity of a function ( f: X to Y ) between topological spaces is defined as the pre-image of every open set in ( Y ) is open in ( X ).

Applications and further study

Topology has many interesting applications beyond pure mathematics. For example, it plays an important role in:

  • Physics: Especially in the fields of general relativity and quantum field theory, topology provides insight into the shape and connectivity of the universe.
  • Data analysis: Through techniques such as topological data analysis (TDA), which can extract meaningful insights from complex datasets.
  • Robotics: In path-planning problems, understanding the connectivity of the configuration space can aid in developing algorithms for navigation.

As a field, topology remains an area of active research, with many subfields such as algebraic topology, differential topology, and geometric topology, each of which explores different aspects and applications.


PHD → 3


U
username
0%
completed in PHD

← Prev (2.5.4)
Wavelets
Next (3.1) →
General Topology

Comments 



Topology

https://www.buddymath.com/phd/3?bmal=en