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 ↓


Metric Spaces


In the world of mathematics, especially in the field of topology, the concept of metric space is fundamental. This concept helps in understanding how we can measure the distance between two points in a more generalized space. Let's learn what metric spaces mean, their importance and some examples to make it easier to understand this concept.

Definition of metric space

A metric space is a set with a function, known as the metric, that calculates the distance between any two elements in the set. More formally, a metric space is a pair (X, d), where:

  • X is a set, and
  • d: X × X → [0, ∞) is a metric, which means that it is a function that takes two elements from X and returns a non-negative real number.

X d satisfy these conditions for all elements a, b, and c:

  1. Non-negativity: d(a, b) ≥ 0 The distance between any two points is non-negative.
  2. Identity of indivisible: d(a, b) = 0 if and only if a = b.
  3. Symmetry: d(a, b) = d(b, a) The distance from a to b is the same as the distance from b to a.
  4. Triangle inequality: d(a, c) ≤ d(a, b) + d(b, c).

These conditions ensure that the notion of distance in metric spaces behaves very similarly to our intuitive understanding of distance in real life.

Visual example

Let us make the concept clear using a simple example - the real number line. The set of real numbers ℝ can be viewed as a metric space. Here, the distance metric d is defined by the absolute difference between two numbers. Consider points a and b on the number line.

A B d(a, b) = |a - b|

Here, d(a, b) = |a - b| is the distance between a and b on the number line, represented by the length of the straight line segment connecting the two points.

Examples of metric spaces

Let's look at some examples so we can understand how diverse and versatile metric spaces can be.

Euclidean space

Probably the most famous example of a metric space is Euclidean space. In n dimensional Euclidean space ℝ n, the distance between two points a = (a 1, a 2, ..., a n) and b = (b 1, b 2, ..., b n) is given by the formula:

d(a, b) = √((b 1 - a 1)² + (b 2 - a 2)² + ... + (b n - a n)²)

This formula is a generalization of the Pythagorean theorem to multi-dimensions.

Discrete metric space

In a discrete metric space, the set X is equipped with a discrete metric d which is defined as:

d(a, b) = 0 if a = b, and 1 otherwise.

This metric is simple, as it defines that points are "near" (zero distance) only if they are similar, otherwise they are "far apart" (unit distance).

Taxicab geometry (Manhattan distance)

In ℝ² with the taxicab metric, the distance between two points a and b is calculated as the sum of the absolute differences of their coordinates:

d(a, b) = |a 1 - b 1| + |a 2 - b 2|

It is called the "Manhattan distance" because it represents how far a person would travel on a grid of streets, as in the layout of the Manhattan borough of New York City.

Why are metric spaces important?

Metric spaces provide the basis for many concepts in topology and analysis. They allow us to talk rigorously about limits, continuity, and convergence, which are core concepts in real analysis and calculus. Furthermore, metric spaces can often be generalized into more abstract topological spaces, providing insight into structures that are not necessarily numerical or geometric.

Properties of metric spaces

Metric spaces have several interesting properties:

Open and closed sets

An open set in a metric space is a set where, for any point within the set, you can travel a small distance in any direction and still remain inside the set. On the other hand, a closed set includes all of its boundary points.

For example, consider the interval (0, 1) on the real number line. This interval is open because you can choose any point between 0 and 1, move a bit, and still remain inside the interval. In contrast, the interval [0, 1] is closed because it contains the endpoints 0 and 1.

Convergence and completeness

A sequence of points in a metric space is said to converge to a point if, eventually, the points of the sequence get arbitrarily close to that point. A metric space is said to be complete if every Cauchy sequence (a sequence where the points get arbitrarily close to each other as the sequence progresses) converges to a point within the space. Real numbers are complete, while rational numbers are not.

Detailed example of a metric space

Consider the set X = {1, 2, 3, 4} with the following metric:

d(a, b) = |a - b|

Here, suppose we calculate some distances:

  • d(1, 2) = |1 - 2| = 1
  • d(1, 3) = |1 - 3| = 2
  • d(2, 4) = |2 - 4| = 2

This set and metric satisfy the criteria for a metric space. Note how each distance obeys the properties of non-negativity, identity of inseparables, symmetry, and triangle inequality.

Conclusion

Metric spaces serve as a fundamental concept connecting various branches of mathematics. They allow for abstract definitions of geometric ideas such as distance between points and provide a framework for understanding spaces that are not immediately intuitive.

This concept is essential in fields such as functional analysis, differential geometry, and topology because it provides clarity and generality. Understanding metric spaces prepares students and mathematicians to delve deeper into more complex and abstract mathematical theories.


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