Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Topology → General Topology ↓


Open and Closed Sets


In the study of topology, one of the fundamental concepts is that of open and closed sets. These ideas form the basis on which more complex structures are built. While open and closed sets may seem abstract, they are in fact closely related to our intuitive understanding of geometry and space.

Intuitive understanding of open sets

Imagine you are in a large park. If we consider this park as a space, then choosing any arbitrary location in this park is the same as choosing a 'point' in space. Now, think of a picnic spot that has no boundary fence. You can move freely in and out of this space without crossing any boundary. This is similar to an open set in topology.

In mathematical terms, a set is said to be open when, roughly speaking, you can move around inside the set without hitting its edge. For every point inside the set, you can find a small neighborhood around that point that is also entirely inside the set.

Visual representation

Open set

In the diagram above, the entire light blue region represents an open set. If you choose any point within the circle, you can find a small neighborhood (perhaps another smaller circle) around it that is still completely contained in the light blue region.

Closed sets: a paradox

Now let's think of the same park, but this time with a fenced boundary. If you are inside this park, you can reach the edge - the fence. This is the same as a closed set. In topology, a set is called closed if it contains all of its boundary points.

Another way to think about closed sets is through their complement relation with open sets. A set is closed if its complement (everything that is not in it) is an open set. This gives rise to one of the important properties of topology – sets can be both open and closed at the same time, or neither.

Visual representation

Closed set

In this diagram, the light coral area is a closed set. If you choose a point on the black edge (boundary), it is still considered part of this closed set.

Formal definitions

Now that we have the intuitive idea out of the way, let’s look at more formal definitions.

Open set definition

A set U in a topological space (X, τ) is called open if U is a member of the topology τ. This means, U ∈ τ.

Mathematically, this can be expressed as: for every point x in U, there exists a neighborhood V around x such that V is completely contained in U

Closed set definition

A set F in a topological space (X, τ) is called closed if its complement, X F, is an open set. That is, X F ∈ τ.

In simple terms, a closed set is one that includes all its limit points (boundary points).

Text examples from Euclidean space

In the familiar Euclidean space R^n, the concepts of open and closed sets correspond to our traditional understanding:

  • In R open interval (a, b) is an example of an open set.
  • closed interval [a, b] is an example of a closed set.
  • empty set and the whole space R^n are both open and closed, known as clopen sets. This may seem contradictory, but remember that these are limit cases in topology.
  • In R^2, an open disk without boundary is an open set.
  • A closed disk, including its boundary circle, is a closed set.

Properties of open sets

Some characterizations of open sets in general topological spaces are:

  • The union of any collection of open sets is open. In other words, if {U_i} are open in X, then their union ∪U_i is also open.
  • The intersection of a finite number of open sets is open. So, if U_1, U_2, ..., U_n are open in X, then U_1 ∩ U_2 ∩ ... ∩ U_n is open.
  • By definition of topology the empty set and the whole space X are always open.

Properties of closed sets

Closed sets have their own nice properties that are twice as many as those of open sets:

  • The intersection of any collection of closed sets is closed. So if {F_i} are closed in X, then ∩F_i is also closed.
  • The union of a finite number of closed sets is closed. Thus, if F_1, F_2, ..., F_n are closed in X, then F_1 ∪ F_2 ∪ ... ∪ F_n is closed.
  • Like open sets, the empty set and the entire space X are also considered to be closed.

Boundary and border points

To further strengthen our understanding, let us delve a little deeper into the concept of limits and limit points.

Limit point

The boundary of a set A in a topological space consists of points that can be reached from both inside and outside A These points do not necessarily lie in A or its complement, but lie on the boundary.

Limit point

A limit point (or accumulation point) of a set A is a point x such that every open neighborhood of x contains at least one point in A distinct from x. Closed sets have all their limit points.

Examples and exercises

Let's work through some examples to apply what we've learned:

  1. Example 1: Consider the set (0, 1) ∪ (2, 3) in R Is it open?
    Answer: Yes, it is open as (0, 1) and (2, 3) both are open intervals in R
  2. Example 2: Consider the set [0, 1] ∪ (2, 3] in R Is it closed?
    Answer: It is not closed because point 2 is a limit point of the set [0, 1] ∪ (2, 3] and is not included in it.
  3. Example 3: Show that the union of an arbitrary collection of open sets is open.
    Solution: Let {U_i}_i be an arbitrary collection of open sets in a topological space X For any point x ∈ ∪U_i, there exists an i such that x ∈ U_i. Since U_i is open, it has a neighborhood completely contained in U_i and hence in ∪U_i.
  4. Exercise: Prove that the intersection of a finite number of closed sets is closed.

Closing thoughts

The concepts of open and closed sets are an important part of topology, a field rich in abstract but applied mathematics. These concepts help define boundaries and are fundamental in understanding limits, continuity, and convergence within topological spaces.

Understanding open and closed sets opens the way for further exploration of deeper concepts such as density, connectedness, and different types of continuum, which in turn paves the way for understanding more sophisticated and beautiful mathematical theories.


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Open and Closed Sets

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