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22:5322:53, 22 March 2026 Topological space (hist | edit) [1,214 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) [1,831 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 Hausdorffness (hist | edit) [3,895 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
21:1921:19, 22 March 2026 Equivalence relation (hist | edit) [862 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 Compactness (hist | edit) [1,038 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
20:5720:57, 22 March 2026 Category of topological spaces (hist | edit) [899 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) [21,471 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

22 March 2026