strangerRidingCaml

Separation Axioms T0, T1, T2, and T3 Separation Axioms T0 Separation Axiom: A topological space \( X \) satisfies the T0 separation axiom if for every pair of distinct points \( x, y \) in \( X \), there exists an open set containing exactly one of them. T1 Separation Axiom: A topological space \( X \) satisfies the T1 separation axiom if for every pair of distinct p..

Topology of Metric Spaces Metric Spaces and Their Topology Metric Space: A metric space is a set \( X \) equipped with a distance function \( d: X \times X \rightarrow \mathbb{R} \) satisfying the following properties for all \( x, y, z \) in \( X \): Non-negativity: \( d(x, y) \geq 0 \) with equality if and only if \( x = y \). Identity of indiscernib..

Continuity and Homeomorphisms Definition of Continuity Between Topological Spaces Continuity: A function \( f: X \rightarrow Y \) between two topological spaces \( (X, \tau_X) \) and \( (Y, \tau_Y) \) is said to be continuous if the preimage of every open set in \( Y \) is open in \( X \). That is, for every open set \( V \) in \( Y \), the set \( f^{-1}(V) \) is open in \( X \). ..

Topological Spaces Definition of a Topology on a Set Topology: A topology on a set \( X \) is a collection \( \tau \) of subsets of \( X \) satisfying the following properties: The empty set and \( X \) itself are in \( \tau \). Any union of sets in \( \tau \) is in \( \tau \). Any finite intersection of sets in \( \tau \) is in \( \tau \). ..

Introduction to Set Theory Basics of Sets, Relations, and Functions Set: A set is a collection of distinct objects, called elements. For example, \( A = \{1, 2, 3\} \) represents a set with elements 1, 2, and 3. Relations: A relation between sets A and B is a subset of their Cartesian product \( A \times B \). Common types of relations include: Bin..

Advanced TopicsSingular Value Decomposition (SVD)The Singular Value Decomposition (SVD) of a matrix $\mathbf{A}$ is a factorization of $\mathbf{A}$ into the product of three matrices:$$\mathbf{A} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^*$$where $\mathbf{U}$ and $\mathbf{V}$ are unitary matrices, and $\mathbf{\Sigma}$ is a diagonal matrix with the singular values of $\mathbf{A}$ on its diagonal.P..