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 ↓


Connectedness


In topology, an interesting concept is “connectedness”. Understanding connectedness helps us understand how points in topological spaces relate to each other and how these spaces are structured in terms of separation. This concept may seem a bit abstract at first, but with some intuitive examples and visualizations, it becomes clear.

Basic definitions

At its core, connectedness refers to the idea that a space cannot be partitioned into two disjoint non-empty open subsets. More formally, a topological space X is called connected if it is not possible to partition X into two non-empty open sets that are disjoint from each other. Here's what that means:

Suppose you have a set X, and you try to partition it into two parts, A and B, where these two parts are:

  • Open in X
  • Disjoint (i.e., A ∩ B = ∅)
  • A ∪ B = X

If you cannot do such a separation, then X is connected.

Visual example

X

In this diagram, consider the space X as a large circle. If you can draw a line that divides this space into two non-overlapping open parts, then X is not connected. If not, it is connected.

Examples of connected spaces

Example 1: Real number line

The real number line, ℝ, is a well-known example of a connected space. Intuitively, you cannot divide the real numbers into two non-empty open intervals without any gaps intersecting. No matter where you try to cut, there will always be a point immediately before or after your cut. Mathematically, there will always be neighborhoods that reconnect the separated parts:

            ℝ = (-∞, a) ∪ (a, ∞)

Here, while the intervals (-∞, a) and (a, ∞) are disjoint, the point a itself ensures associativity.

Example 2: A circle

Another example is a circle. If you take a topological space that is shaped like a circle, it is connected. You cannot separate a circle into two open paths unless it eventually crosses or touches a boundary that brings the circles back together.

Imagine trying to draw a line that divides the circle into two disjoint open paths; this is not possible without touching or joining the boundary.

Disconnected spaces

A space that is not connected is called disjoint. If a space can be partitioned into two or more disjoint non-empty open subsets, then it is called disjoint. Let's look at some examples:

Example 1: Two distinct points

Consider the set {1, 2}. This set is clearly disjoint, since there is no way to go continuously from point 1 to point 2. There are two disjoint open sets that cover the space:

            A = {1}, B = {2}

Here, both A and B are open, and since they are separate, they separate the space.

Properties of connected spaces

Intermediate value theorem

An important property of connected spaces concerns the Intermediate Value Theorem. This theorem states that if a function f is continuous on a connected space X and a and b are two values within the range of f, then f takes on every value between a and b.

Path affiliation

Path-connectedness is a related concept, where a space is called path-connected if any two points in that space can be joined by a continuous path. This is slightly stronger than connectedness because all path-connected spaces are connected, but not all connected spaces are path-connected.

Path association visualization example

In this example, consider the path that connects two points on the space. You can see that the space is connected because it is possible to find a connected path without separation.

Connected components

Every topological space can be partitioned into its maximal connected subsets, known as its connected components. These components are disjoint and their union forms the whole space.

Example of connected components

Consider a space consisting of two disjoint circles. Here, each circle is itself a connected component, since there is no continuous path between two circles.

The importance of engagement

The concept of connectivity simplifies the analysis of topological spaces in various domains such as complex analysis, algebraic topology, and more. In complex analysis, connectivity helps define domains for analytic functions. Additionally, in physics, connectivity can explain how different regions of space and space-time can be naturally connected.

Conclusion

Connectedness is a fundamental concept in topology that helps us understand how topological spaces can interact and become fragmented. By understanding this concept, we gain insight into the behavior of various mathematical and physical systems. Understanding whether a space is connected or disconnected helps us predict how the system will behave under various transformations.


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