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


Topology is a fascinating field of mathematics that advances the study of geometry. Unlike traditional geometry, which deals with size, distance, and exact shape, topology is about the fundamental nature of space. It is about understanding which properties of shapes and spaces remain preserved through stretching, twisting, and bending, without tearing or chipping. Topology allows us to explore the properties of objects that remain unchanged, no matter how the object changes.

Basic concepts in topology

To understand topology, let's understand some essential concepts:

Topological spaces

A topological space is the fundamental subject of study in topology. It is a set of points around which there is a neighborhood where certain rules are satisfied. Formally, a set X forms a topological space with a collection of subsets T (called the topology) if:

  • The empty set ∅ and X itself belong to T
  • The union of any collection of sets in T is also in T
  • The intersection of any finite number of sets in T also lies in T

Think of it as a set of objects, and the topology defines which objects we can group together and consider 'open'.

Open and closed sets

Open sets are a cornerstone of topology. Intuitively, an open set is a set where, no matter which point you choose, you can move a little bit in any direction without leaving the set. For example, in a two-dimensional plane, a circle without boundary is an open set. A closed set is the complement of an open set with respect to the whole space. An example is a circle that includes its boundary.

Continuity and homeomorphisms

One might wonder how topology handles the idea of a function. A function between two topological spaces is continuous if the preimage of every open set is open. A powerful concept in topology is homeomorphism. A homeomorphism is a bi-continuous function; in other words, if two spaces are homeomorphic, then there exists a continuous function with a continuous inverse connecting them. This means that the two spaces are topologically the same, even if they look geometrically different.

Visualization of topological concepts

Topology is often easier to understand visually. Let's look at some common examples:

Circle and ellipse

Consider a circle and an ellipse. Geometrically, they are different. However, in topology, they are considered equivalent because you can stretch or compress a circle into an ellipse without any cutting or gluing. They are both simple closed curves.

In the above figure, the circle on the left and the ellipse on the right are topologically equivalent.

Coffee cup and donut

One of the most famous and classic examples in topology is the similarity between a coffee cup and a donut (or torus). Although they look different, a coffee cup with a handle can be deformed into a torus shape without cutting it, which makes them homeomorphic.

In the above illustration, the torus on the right shows a donut shape that is topographically similar to a coffee cup with a handle.

Mobius strip

The Mobius strip is another interesting object in topology. It is a surface with only one side and a boundary component. To create a Mobius strip, take a strip of paper, fold it in half and then join the ends together. The fascinating part about the Mobius strip is that if you start drawing a line in the middle, you will eventually return to your starting point in the opposite direction.

The path above gives an indication of how a Möbius strip might be viewed, and also shows a twist.

Important theorems and concepts

Compactness

A topological space is compact if every open cover has a finite subcover. In simple terms, compact spaces allow us to make broad assumptions about their behavior because they do not extend to infinity, and functions on compact spaces often have nice properties. An essential part of compactness is that it allows the generalization of many theorems in calculus, including extensions of the idea of limits and continuity.

Connectedness

A space is called connected if it cannot be partitioned into two disjoint open sets. Informally, a connected space is "all in one piece". For example, a line or a circle is connected, but a pair of disjoint circles is not connected.

The above example shows two disjoint circles, which is a disjoint space.

Topological invariants

Euler characteristic

The Euler characteristic is a number that describes the shape or structure of a topological space, no matter how it is bent. It is defined for a solid shape as follows:

Euler characteristic = V – E + F

where V is the number of vertices, E is the number of edges, and F is the number of faces. For a simple polyhedron such as a cube, the Euler characteristic is 2.

Fundamental group

The fundamental group is a concept that helps classify spaces based on path-connectivity. It gives information about loops in a space. For example, the fundamental group of a circle is infinite, which shows that you can wrap the circle around itself indefinitely.

Applications of topology

The applications of topology also extend beyond pure mathematics. It provides tools and concepts applicable in a variety of fields, including data analysis, physics, and computer graphics.

Data analysis

In data analysis, topological methods can identify patterns and structures in data sets. For example, persistent symmetry is used in topological data analysis to study the shape of data.

Theoretical physics

In physics, topological concepts are useful in quantum mechanics and general relativity. The topology of the universe, the study of space-time and topological insulators are some of the areas that rely heavily on topological ideas.

Conclusion

Topology is a broad and deep field of mathematics that goes beyond traditional geometry. It invites us to look beyond the surface and consider underlying properties that are not readily apparent. Whether studying space, understanding connectivity, or applying its principles to modern technology and science, topology provides a unique lens through which we can explore the world of mathematics and beyond.


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