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 ↓


Topological Spaces


Topology is a fascinating field of mathematics that extends concepts of geometry and other branches. At its core lies the idea of continuity, location, and the shape of spaces, which can differ significantly from shapes in Euclidean geometry. One of the key concepts in topology is the topological space, a set with a structure that allows the definition of concepts such as convergence, continuity, and limit.

What is a topological space?

A topological space is a set equipped with a topology, a collection of subsets that satisfy certain properties. These properties ensure that the subsets can serve as "open sets" of the space, generalizing the notion of open intervals in the real number topology.

Definition

A topological space is defined as a pair ( (X, tau) ) where:

  • ( X ) is a set.
  • ( tau ) (called the topology on ( X )) is a collection of subsets of ( X ) that satisfy the following properties:
    • The empty set ( emptyset ) and ( X ) itself are in ( tau ).
    • The union of any collection of sets in ( tau ) is also in ( tau ).
    • The intersection of any finite number of sets in ( tau ) also occurs in ( tau ).

The sets in ( tau ) are called open sets . The elements of ( X ) are called points .

Examples of topological spaces

1. Discrete topology

Consider a set ( X ) consisting of three elements: ( X = {a, b, c} ). The discrete topology on ( X ) is the power set of ( X ), which means that every subset of ( X ) is open:

( tau = {emptyset, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} )

Every subset of ( X ) is open in the discrete topology. This implies maximum flexibility, since any point can be isolated.

2. Trivial topology

In the trivial topology, only the empty set and the whole set are open sets. For the same set ( X = {a, b, c} ), the trivial topology is:

( tau = {emptyset, X} )

In this topology the openness is minimal. No point is isolated.

3. The standard topology on the real line

Consider the set ( mathbb{R} ) of all real numbers. The standard topology on ( mathbb{R} ) is formed by open intervals:

( tau = {(a, b) mid a, b in mathbb{R}, a < b} )

This means that any open interval and any union of open intervals are open sets in ( mathbb{R} ).

Visualization of topological spaces

The understanding of topological spaces can be enhanced through visual examples.

Visual example: Discrete topology

A B C

This reflects the discrete topology on ( X = {a, b, c} ). Every point can stand alone as an open set, since any subset of the space is open.

Visual example: Union of intervals

(a, b) (CD)

This represents the union of open intervals in the real line, an open set in the standard topology.

Basic concepts in topology

Open and closed sets

Open sets are already defined through topology. A closed set is a set whose complement is open. For example, in the real numbers, the closed interval ( [a, b] ) is not open, but its complement ( (-infty, a) cup (b, infty) ) is open.

Interior, closure, and boundary

Given a topological space ( (X, tau) ) and a subset ( A subset X ):

  • The interior of ( A ) is the largest open set contained in ( A ).
  • The closure of ( A ) is the smallest closed set containing ( A ).
  • The boundary of ( A ) is the specified difference between the closure and the interior.

For example, consider the subset ( A = (0, 1) ) in the real numbers:

  • The interior of ( A ) is ( (0, 1) ).
  • The termination is ( [0, 1] ).
  • The range is ( {0, 1} ).

Continuous work

Continuum in topology generalizes the notion of continuum from calculus.

Definition of continuity

A function ( f: (X, tau_X) to (Y, tau_Y) ) between two topological spaces is continuous if for every open set ( V ) in ( Y ), the preimage ( f^{-1}(V) ) is open in ( X ).

This framework allows for greater generality than real analysis and connects naturally with many physical phenomena involving continuous deformation.

Continuous task visualization example

F

It represents an imaginary continuous function from a circular structure at one location to a rectangular shape at another, indicating deformation and continuum.

Compactness and connectedness

Compactness

A subset ( A ) of a topological space ( (X, tau) ) is compact if every open cover of ( A ) has a finite subcover. This generalizes the notion of closed and bounded subsets to the real numbers.

For example, closed intervals like ( [a, b] ) in ( mathbb{R} ) are compact in the standard topology.

Connectedness

A space is connected if it cannot be divided into two non-empty disjoint open subsets. For example, the interval ( (0, 1) ) is connected because there is no way to divide it into two open and disjoint parts.

Visual example

Disconnected Points Connected path

This illustration shows connected paths versus unconnected points, and illustrates the concept of connectedness in topology.

Conclusion

Topological spaces form the backbone of topology, underpinning many concepts and applications. By understanding open and closed sets, continuous functions, and properties such as compactness and connectedness, a foundational gateway to more advanced topics in mathematics is opened. Through these explorations, mathematics delves deeper into areas that are not bound by regular geometric constraints, allowing for the investigation of more abstract and complex constructions in a coherent and structured way.


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Topological Spaces

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