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15:3715:37, 14 May 2026 Seifert-Van Kampen theorem (hist | edit) [1,001 bytes] InfernalAtom683 (talk | contribs) (Created page with "== Statement == Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps: <math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math> then the following d...") Tag: Visual edit: Switched

21:5721:57, 29 April 2026 Commutator (hist | edit) [4,305 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''commutator''' is an algebraic expression that measures the failure of two elements to commute. It occurs throughout abstract algebra, particularly in group theory, ring theory, and linear algebra. If two elements commute, their commutator is trivial. More generally, the commutator describes the obstruction to exchanging the order of two operations. Commutators are fundamental in the study of noncommutative structures and in the construction of i...")

16:1816:18, 5 April 2026 Quotient group (hist | edit) [985 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''quotient group''' is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. == Definitions == Let <math>G</math> be a group and <math>N\trianglelefteq G</math> a normal subgroup. === Definition via cosets === The quotient group <math>G/N</math> is the set of left cosets <math display="block">G/N:=\{gN\mid g\in G\}</math>with group operation <math>(gN)(hN):=(gh)N.</math> === Defi...") Tag: Visual edit

19:1819:18, 4 April 2026 First isomorphism theorem (hist | edit) [3,083 bytes] InfernalAtom683 (talk | contribs) (Created page with "The '''first isomorphism theorem''' is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeo...") Tag: Visual edit

15:2915:29, 28 March 2026 Darboux integral (hist | edit) [2,841 bytes] InfernalAtom683 (talk | contribs) (Created page with "The '''Darboux integral''' is a formulation of integration in real analysis defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the Riemann integral. == Definition == === Darboux sums === Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block"> P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b </math> be a partition of the interval...") Tag: Visual edit: Switched

21:3921:39, 24 March 2026 Computation (hist | edit) [16,057 bytes] Michaelihc (talk | contribs) (Created page (draft)) Tag: Visual edit
12:2912:29, 24 March 2026 Locally Euclidean space (hist | edit) [1,562 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''locally Euclidean space''' is a topological space that resembles a Euclidean space locally. Specifically, every point in a locally Euclidean space has an open neighborhood that is homeomorphic to an open subset in <math>\mathbb{R}^n</math>. The concept of locally Euclidean space is a central object in the definition of topological manifold. == Definition == A topological space <math>X</math> is locally Euclidean of dimension <math>n</math>, if for every...") Tag: Visual edit

22:5322:53, 22 March 2026 Topological space (hist | edit) [3,088 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''topological space''' is a fundamental mathematical structure that generalizes the concept of geometrical spaces and continuity. A topological space is equipped with a collection of open sets, capturing the intuitive idea of "nearness" without necessarily defining a metric. Topological spaces are the objects of study in general topology. == Definition == An ordered pair <math>(X, \tau)</math> is a topological space on set <math>X</math>, if <math...") Tag: Visual edit
21:2121:21, 22 March 2026 Homotopy (hist | edit) [2,652 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''homotopy''' is a continuous deformation between two continuous functions from one topological space to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic. Intuitively, a homotopy is the continuous transformation of...") Tag: Visual edit
21:2021:20, 22 March 2026 Hausdorff space (hist | edit) [4,187 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''Hausdorff space''' (or '''<math>T_2</math> space''') is a type of topological space in which points can be "cleanly separated" by neighborhoods. Specifically, for any two distinct points, there exist disjoint open sets containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist. == Definitions == A topological space <math>(X,\tau)</math> is Hausdorff, if for any two points <math>x,y\in X</math...") Tag: Visual edit originally created as "Hausdorffness"
21:1921:19, 22 March 2026 Equivalence relation (hist | edit) [1,860 bytes] InfernalAtom683 (talk | contribs) (Created page with "An '''equivalence relation''' is a binary relation on a set that groups elements into categories<ref>Not to be confused with category in category theory.</ref> in which all members are considered "equivalent" under some criterion. == Definition == A relation <math>\sim</math> on set <math>X</math> is a equivalence relation if it satisfies the following properties: * '''Reflexivity''': <math>\forall x\in X</math>, <math>x\sim x</math>. * '''Symmetry'...") Tag: Visual edit
20:5820:58, 22 March 2026 Compact space (hist | edit) [1,397 bytes] InfernalAtom683 (talk | contribs) (Created page with "A '''compact''' topological space is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental topological properties in analysis and topology. Intuitively, compactness can be understood as a generalization of being "closed and bounded"....") Tag: Visual edit originally created as "Compactness"
20:5720:57, 22 March 2026 Category of topological spaces (hist | edit) [2,447 bytes] InfernalAtom683 (talk | contribs) (Created page with "The '''category of topological spaces''', denoted <math>\mathsf{Top}</math> or <math>\mathbf{Top}</math>, is the category whose objects are topological spaces and whose morphisms are continuous functions. == Definition == The category <math>\mathsf{Top}</math> consists of: * <math>\operatorname{ob}(\mathsf{Top})</math> consists of all topological spaces, * <math>\operatorname{mor}(\mathsf{Top})</math> consi...") Tag: Visual edit
20:4220:42, 22 March 2026 Homeomorphism (hist | edit) [27,340 bytes] InfernalAtom683 (talk | contribs) (Created page with "thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously. A '''homeomorphism''' is a special type of function between two topological spaces, that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a continuous bijective function whose inverse function is also continuous. Homeomorphisms are the Isomorp...") Tag: Visual edit: Switched

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