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 ↓


Continuity and Homeomorphisms


Topology is a fascinating field of mathematics that goes beyond local shapes and measures to deal with more abstract concepts such as continuity and equivalence between spaces. Two key topics in topology that are fundamental for undergraduate mathematics students to understand are continuity and homeomorphism. Let's understand these concepts carefully using simple language and examples.

Understanding continuity

Continuity is a concept most students encounter early in calculus, but it becomes much more subtle in topology. The basic idea of continuity concerns how functions and their spaces relate.

Basic definition

In topology, a function f: X rightarrow Y between two topological spaces X and Y is continuous if for every open set V subseteq Y, the preimage f^{-1}(V) is an open set in X

F : X → Y Continuous if, ∀ open V ⊆ Y, f⁻¹(V) ⊆ X is open.

The idea of continuity

The idea of joining parts of one space to another without 'tearing' or 'pasting' them can be explained as follows:

X Y

Here, the function is mapping the open set represented by the light blue rectangle in space X to the green rectangle in space Y

Examples of continuous functions

Let's look at some examples to strengthen our understanding of continuous operations:

  • Consider the function f: ℝ rightarrow ℝ defined by f(x) = 2x. This function is continuous because any open interval in ℝ has a pre-image under f that is also an open interval.
  • The function g: S^1 rightarrow ℝ^2 that maps each point on the circle to its coordinates in two-dimensional space is continuous. It does not 'break' the circle, but rather maps each point smoothly.

Homeomorphisms: twists on continuity

A homeomorphism is a type of function that represents a stronger form of equivalence between topological spaces. Two spaces are homeomorphic if there exists a continuous function between them with a continuous inverse.

Definition and properties

A function f: X rightarrow Y is a homeomorphism if:

  • f is binary (one-to-one and integer).
  • f is continuous.
  • The inverse function f^{-1}: Y rightarrow X is continuous.
f : X → Y is a homeomorphism ⇔ (f is bijective ∧ f is continuous ∧ f⁻¹ is continuous)

Homeomorphic spaces are often informally said to be topologically "identical". They can be deformed into one another without cutting or pasting.

Examples of homeomorphisms

Let's explore some examples of homeomorphic spaces:

  • The unit circle S^1 is isomorphic to any ellipse, because you can continuously stretch and compress a circle to make an ellipse, and vice versa.
  • The open interval (0, 1) is isomorphic to the real number line ℝ. Although this seems surprising, it can be represented with functions such as the tangent function that extend one over the other.

Visualizing homeomorphisms

Visually, homeomorphisms can be represented by a continuous deformation from one shape to another:

S¹ (circle) Oval

In this illustration, the circle continuously transforms into an ellipse. This is a visual analogy of what a homeomorphism between these spaces achieves.

Importance of homeomorphisms in topology

Homeomorphisms are important in topology because they allow us to classify spaces according to their essential shape or 'topological' nature. If two spaces are homeomorphic, we consider them to be topologically the same, even if they look different at first glance.

Why care about homeomorphisms?

Understanding the concept of homeomorphism has several important implications:

  • They help us understand and simplify complex spaces by classifying them into equivalence classes.
  • Many space invariants preserved through homeomorphisms help to identify these equivalence classes.
  • Topological invariants, such as valency and compactness, remain unchanged under homeomorphisms.

Closing thoughts

Continuity and homeomorphism are the backbone of many sophisticated ideas in topology. With continuity, we build a bridge between spaces that preserves the structure of openness, while homeomorphisms are bridges that create complete equivalence in the language of topology. As you dive deeper into the world of topology, remember that these concepts will be your guiding tools to understand the intrinsic nature of spaces.


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Continuity and Homeomorphisms

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