12.1 Set theory in topology and analysis

Set theory forms the foundation for topology and analysis, providing tools to study spaces and functions. In topology, sets are used to define open and closed sets, continuity, and topological properties.

In analysis, set theory helps define measures, integrals, and function spaces. These concepts are crucial for understanding limits, convergence, and advanced mathematical structures used in various fields.

Topological Spaces

Defining Topological Spaces and Open Sets

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• Topological space defined as a set $X$ together with a collection of subsets of $X$ called open sets
• Open sets satisfy the following axioms:
  • The empty set $\emptyset$ and the entire set $X$ are open
  • The union of any collection of open sets is open
  • The intersection of a finite number of open sets is open
• Examples of topological spaces include:
  • Euclidean space $\mathbb{R}^n$ with the standard topology (open sets are unions of open balls)
  • Discrete topology on any set $X$ where every subset of $X$ is open

Closed Sets and Continuous Functions

• Closed set defined as the complement of an open set
  • A set $A \subseteq X$ is closed if and only if its complement $X \setminus A$ is open
• Continuous function between topological spaces preserves the topological structure
  • A function $f: X \to Y$ is continuous if the preimage of every open set in $Y$ is open in $X$
  • Intuitively, continuous functions map nearby points in the domain to nearby points in the codomain
• Examples of continuous functions:
  • Identity function $f(x) = x$ is always continuous
  • Constant functions $f(x) = c$ are always continuous

Metric Spaces

Defining Metric Spaces

• Metric space defined as a set $X$ together with a distance function (metric) $d: X \times X \to [0, \infty)$ satisfying:
  • Non-negativity: $d(x, y) \geq 0$ for all $x, y \in X$
  • Identity of indiscernibles: $d(x, y) = 0$ if and only if $x = y$
  • Symmetry: $d(x, y) = d(y, x)$ for all $x, y \in X$
  • Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in X$
• Examples of metric spaces:
  • Euclidean space $\mathbb{R}^n$ with the Euclidean metric $d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$
  • Discrete metric space where $d(x, y) = 1$ if $x \neq y$ and $d(x, x) = 0$

Compactness and Connectedness in Metric Spaces

• Compactness is a generalization of the notion of a closed and bounded subset of Euclidean space
  • A metric space is compact if every open cover has a finite subcover
  • Compact sets are closed and bounded, but the converse is not always true (e.g., in infinite-dimensional spaces)
• Connectedness captures the idea of a space being in one piece
  • A metric space is connected if it cannot be written as the union of two disjoint, non-empty open sets
  • Path-connectedness implies connectedness, but the converse is not always true (e.g., the topologist's sine curve)

Measure Theory

Foundations of Measure Theory

• Measure theory extends the notion of length, area, and volume to more general sets
• Measure defined as a function $\mu$ that assigns a non-negative real number or $\infty$ to subsets of a set $X$, satisfying:
  • Non-negativity: $\mu(A) \geq 0$ for all $A \subseteq X$
  • Null empty set: $\mu(\emptyset) = 0$
  • Countable additivity: For a countable collection of disjoint sets $\{A_i\}_{i=1}^\infty$, $\mu(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \mu(A_i)$
• Examples of measures include:
  • Counting measure on a countable set, where $\mu(A)$ is the number of elements in $A$
  • Probability measures, where $\mu(X) = 1$

Lebesgue Measure and Borel Sets

• Lebesgue measure is an extension of the classical notions of length, area, and volume to a larger class of sets
  • Lebesgue measure on $\mathbb{R}^n$ assigns the usual volume to "nice" sets (e.g., rectangles, balls) and is extended to more complicated sets using the concept of Lebesgue measurability
• Borel sets form the smallest collection of subsets of a topological space that contains all open sets and is closed under countable unions and intersections
  • In $\mathbb{R}^n$, Borel sets include open sets, closed sets, and many other sets encountered in analysis
  • Lebesgue measure is defined on the Borel $\sigma$-algebra, which contains all Borel sets
• Examples of Lebesgue measurable sets:
  • All Borel sets, including open sets, closed sets, and countable sets
  • Sets of Lebesgue measure zero, such as the Cantor set