Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Topology ↓


Compactness


Compactness is a concept in topology, a branch of mathematics that deals with the properties of space that are preserved under continuous transformations. It is an important concept with a wide range of applications within topology and beyond.

Intuitively, compactness can be thought of as a finite cover property. To take a deeper look at this, we begin by understanding how this notion is formalized in topology.

Definitions and basics

In topology, a space is said to be compact if every open cover of the space has a finite subcover. Let's understand these terms:

  • Topological Space: A set X and a collection of open sets adjacent to it that satisfy some axioms.
  • Open cover: A collection of open sets whose union contains a space X
  • Finite subcover: A finite selection of open sets from an open cover that still covers the whole space.

Mathematically, let's use X to denote a topological space. An open cover of X is a collection {U α } α ∈ A such that ∪ U α = X.

There exists a finite subset {U i1, U i2, ..., U in} such that: ∪ U ij = X, for j in {i1, i2, ..., in}.

Examples of compact spaces

The simplest example of a compact space is a closed interval in the real numbers, say [a, b].

Consider the real line ℝ. A closed and bounded subset [a, b] of ℝ is compact. This is a consequence of the Heine-Borel theorem, which states that a subset of ℝ n is compact if and only if it is closed and bounded.

Example: Let [0, 1] be a closed interval. Consider an open cover consisting of open sets (0, 1/n) for n = 1, 2, 3, ... A finite subcover of this would be { (0, 1/2), (0, 1/3) }, etc., that cover [0, 1].
A B

In this diagram, you see the closed interval [a, b], which is compact because any open cover of this interval will have a finite subcover that ensures that the whole interval is covered.

Counterexample: non-compact space

To better understand compactness, it is also useful to consider spaces that are not compact.

Consider the open interval (0, 1) in the real numbers. It will have an open cover { (0, 1 - 1/n) | n ∈ ℕ }. Here, there is no finite subcover covering the whole interval, which shows that (0, 1) is not compact.

Example: (0, 1) is not compact. An open cover: { (0, 1), (0, 0.9), (0, 0.99), ... } It cannot be finitely covered.
0 1

In this diagram, the interval (0, 1) is depicted dashed and open at the ends to demonstrate that it cannot be finitely covered with any collection of open sets that falls entirely within (0, 1).

Properties of compact spaces

Compact spaces exhibit several useful and important properties:

  • Closure and boundedness: In Euclidean spaces, compactness is related to closedness and boundedness (Heine–Borel theorem).
  • The continuous image of a compact space is compact: if a continuous function maps a compact space into another space, then the image is also compact.
  • Finite Intersection Property: A family of closed sets in a compact space with the finite intersection property has a nonempty intersection.

Brevity and continuity

A fascinating aspect of compactness is its interaction with continuous functions. If f: X → Y is a continuous function and X is compact, then f(X) is compact in Y

Let X be compact and f: X → Y continuous. Then, f(X) is compact in Y.

It may help to visualize this:

X f(x)

Here, the circle represents the compact space X, and its continuous mapping into another space is also denoted as compact.

Finite intersection property

Compact spaces have the finite intersection property, which means that if you have a collection of closed sets in a compact space such that every finite intersection of these sets is nonempty, then the whole collection will also have a nonempty intersection.

Let {F α } be a collection of closed sets in X (where X is compact). If ∩ F j ≠ ∅ for every finite subset {F j }, then ∩ F α ≠ ∅.

Applications of compactness

Compactness is useful in various areas of mathematics. One of its applications is in optimization problems, where compactness ensures the existence of a maximum or minimum of a continuous function.

Mathematically it is expressed as:

Let f: X → ℝ be a continuous function on a compact space X. Then, f attains a maximum and a minimum on X.

This can be seen from the following example, where a continuous function on a compact interval attains a maximum and a minimum:

Min Maximum

The blue curve represents the function, while the green and red circles indicate where it attains a minimum and maximum, respectively, within the compact interval.

Conclusion

Compactness is a central concept in topology that gives rise to many powerful theorems and applications. By ensuring that infinite behaviors can be controlled or collected in finite means, compactness enables mathematicians and scientists to draw important conclusions in a variety of mathematical fields.


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