Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Topology ↓


General Topology


General topology, sometimes referred to as point-set topology, is a branch of mathematics that studies the properties of spaces that are topological in nature. In essence, it deals with concepts such as continuity, compactness, and connectedness. These concepts extend ideas from geometry and analysis to a more general set of spaces.

Understanding topological spaces

A topological space is a set of points, in which for each point there is a set of neighborhoods that satisfy a set of axioms relating points and neighborhoods.

Definition: A topology on a set X is a collection T of subsets of X, satisfying: 1. Both the empty set and X are elements of T. 2. Any union of elements of T is an element of T. 3. Any finite intersection of elements of T is an element of T. The pair (X, T) is referred to as a topological space.

For example, consider a set X = {a, b, c} with a topology T = {{}, {a}, {a, b}, X}

This topology on X satisfies all the necessary conditions:

  • The empty set {} and the set X are in T
  • The union of any elements of T, for example, {a} ∪ {a, b} = {a, b}, is also in T
  • The intersection of any element of T, for example, {a} ∩ {a, b} = {a}, is also in T

Open and closed sets

In topology, the distinction between open and closed sets is fundamental. An open set is a member of the topology.

Conversely, if a set is closed then its complement is open. It is important to note that a set can be both open and closed, or neither.

Foundations of topology

The basis for a topology on a set X is the collection B of subsets of X (called basis elements) such that every open set is a union of basis elements. This concept simplifies the definition of topology by focusing on basic continuous building blocks.

Definition: A collection B of subsets of X is called a basis for a topology on X if: 1. For each x in X, there is at least one basis element B containing x. 2. If x belongs to the intersection of two basis elements B1 and B2, there exists a basis element B3 such that x belongs to B3 and B3 is contained in B1 ∩ B2.

Continuous work

Continuity is a fundamental concept in topology. A function between two topological spaces is continuous if, intuitively, it takes points "close to each other" in the domain to points "close to each other" in the range.

Definition: A function f: (X, T) → (Y, U) is continuous if for every open set V in U, the preimage f -1 (V) is open in T.

For example, consider the function f: ℝ → ℝ defined by f(x) = x^2. This function is continuous under the standard topology on ℝ.

Homeomorphisms

A function is a homeomorphism if it is a continuous bijection with a continuous inverse. Homeomorphisms are important because they define that two topological spaces are essentially the same, or "topologically equivalent."

Compactness

A space is compact if every open cover has a finite subcover. This property generalizes the notion of a space being closed and bounded in Euclidean space.

Definition: A topological space (X, T) is compact if for every collection of open sets {Ua} such that the union of Ua covers X, there exists a finite subcollection that covers X.

An example of a compact space is the closed interval [0,1] over the real numbers with the usual topology.

Connectedness

A topological space is connected if it cannot be partitioned into two disjoint nonempty open sets. In simple terms, the space is in a piece.

Definition: A topological space (X, T) is connected if there do not exist two disjoint non-empty open sets U and V such that X = U ∪ V.

The real number line ℝ under the standard topology is an example of a connected space.

Visual example

Let's use some visual examples to explain the concepts of open sets and continuum:

Open set image Continuous maps

Why general topology?

General topology provides essential information about the nature of space and functions. These ideas have applications in a variety of mathematical disciplines, including differential geometry, algebraic topology, and functional analysis.

Final thoughts

As you move further into the world of general topology, understanding basic concepts such as open and closed sets, bases for topology, continuity, compactness, and connectedness will form the foundation on which more complex ideas are built. Engaging with these at both a theoretical and practical level will enable you to appreciate the richness and depth of topological studies.


Graduate → 3.1


U
username
0%
completed in Graduate

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

Comments 



General Topology

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