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 → Real Analysis → Metric Spaces ↓


Open and Closed Sets


In the field of real analysis, an essential concept is that of open and closed sets, specifically within metric spaces. Before delving into the main discussion, it is important to understand what a metric space is. A metric space is a set equipped with a metric, which is a function that defines the distance between each pair of points in the set. A concrete and common example of a metric space is the set of real numbers whose usual distance is defined as:

d(x, y) = |x - y|

where |x - y| is the absolute value of the difference between x and y. However, metric spaces can be much more abstract and include spaces of functions, multidimensional points, and much more.

Understanding open sets

An open set in a metric space is an intuitive concept that is similar to the idea of an interval that does not contain its endpoints. To formally define open sets:

A set U in a metric space (X, d) is called open if for every point x in U there exists a ball of radius r > 0 such that the ball B(x, r) is contained in U Here, B(x, r) denotes the set of all points y in X such that d(x, y) < r.

This might seem a bit abstract at first glance, but let’s understand it with a simple example.

Example of an open set on the real line

Consider the interval (0, 1) on the real number line. Let us check whether it is an open set. Choose any point x in the interval such that 0.5. We have to find a radius r such that all points within a distance r of x still lie inside the interval (0, 1).

Choose r = 0.1. The ball B(0.5, 0.1) is the set of all points y such that |y - 0.5| < 0.1, which is the interval (0.4, 0.6). Clearly, (0.4, 0.6) is completely contained within the interval (0, 1), which shows that (0, 1) is indeed an open set.

+--------------------------+ 0 0.4 0.5 0.6 1

Visualization of open sets

X Y

In the visualization above, each circle around x and y points represents an open ball that is entirely contained in the interval. The line represents the entire real number line, while the segment between the circles represents the open interval.

Properties of open sets

There are several important properties of open sets:

  • The union of any collection of open sets is open.
  • The intersection of a finite number of open sets is open.
  • The whole space X and the empty set ∅ are both open.

Understanding closed sets

The complements of open sets are closed sets. The idea of closed sets is linked to the concept of limits and accumulation points. Formally:

A set C in a metric space (X, d) is closed if it contains all its limit points. Equivalently, C is closed if the complement of C in X, denoted by X C, is open.

Example of a closed set on the real line

Consider the closed interval [0, 1]. Does it contain all its boundary points? Let us take a point x = 0 on the boundary. Outside this point there is no other point y within the interval, so that any sequence converging to 0 lies outside [0, 1]. Every subsequence lies within the boundary [0, 1].

x ∈ [0, 1] such that x_n → x

This shows that [0, 1] is closed since it contains all the boundary points.

Visualization of closed sets

0 1

In the above visualization, the end points are included, which shows the closed nature of [0, 1]. The black line represents the real number line and the filled circles represent the included end points.

Properties of closed sets

Closed sets have specific properties that form a fundamental framework for analysis:

  • The intersection of any collection of closed sets is closed.
  • The union of a finite number of closed sets is closed.
  • The whole space X and the empty set ∅ are both closed.

Beyond open and close: limit points and boundary points

For a deeper understanding of open and closed sets it is useful to explore related concepts such as limit points and boundary points.

Limit point

A limit point of a set S is a point x that has at least one point of S different from x in every neighborhood. This helps to identify the closure of S, which is the smallest closed set that contains S

Limit point

The limit point of a set S is the point b for which every neighbourhood intersects both S and its complement. It serves as the interface between the set and the space beyond it.

Conclusion

Open and closed sets form the fundamental building blocks in the study of metric and topological spaces. Open sets are intimately connected to the concepts of continuity and convergence, with real-world parallels in the form of open intervals. Closed sets harbor limit points and are fundamental in defining completeness and compactness. Understanding one often aids in understanding the other, and together, these sets help illustrate the rich tapestry of relations in analytic spaces.

From intervals on the real line to generalized higher-dimensional spaces, the ideas of open and closed sets surface again and again, uncovering structure and meaning in mathematical landscapes. Armed with these ideas, more complex concepts in analysis become accessible, setting the stage for deeper exploration and discovery.


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Open and Closed Sets

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