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Connected space
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{{Short description|Topological space that is connected}} {{Use American English|date = March 2019}} {{multiple image <!-- Essential parameters --> | align = right<!-- left/right/center --> | direction = vertical<!-- horizontal/vertical --> | width = 200<!-- Digits only; no "px" suffix, please --> <!-- Image 1 --> | image1 = Simply connected, connected, and non-connected spaces.svg<!-- Filename only; no "File:" or "Image:" prefix, please --> | width1 = | alt1 = | caption1 = From top to bottom: red space <em>A</em>, pink space <em>B</em>, yellow space <em>C</em> and orange space <em>D</em> are all <em>connected spaces</em>, whereas green space <em>E</em> (made of [[subset]]s E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, and E<sub>4</sub>) is <em>disconnected</em>. Furthermore, <em>A</em> and <em>B</em> are also [[Simply connected space|simply connected]] ([[Genus (mathematics)|genus]] 0), while <em>C</em> and <em>D</em> are not: <em>C</em> has genus 1 and <em>D</em> has genus 4. <!-- Image 2 --> | image2 = | width2 = | alt2 = | caption2 = <!-- up to |image10 is accepted --> <!-- Extra parameters --> | header = Connected and disconnected subspaces of '''R'''Β² | header_align = <!-- left/right/center --> | header_background = | footer = | footer_align = <!-- left/right/center --> | footer_background = | background color = }} In [[topology]] and related branches of [[mathematics]], a '''connected space''' is a [[topological space]] that cannot be represented as the [[union (set theory)|union]] of two or more [[disjoint set|disjoint]] [[Empty set|non-empty]] [[open (topology)|open subsets]]. Connectedness is one of the principal [[topological properties]] that distinguish topological spaces. A subset of a topological space <math>X</math> is a <em>{{visible anchor|connected set}}</em> if it is a connected space when viewed as a [[Subspace topology|subspace]] of <math>X</math>. Some related but stronger conditions are [[#Path connectedness|path connected]], [[Simply connected space|simply connected]], and [[N-connected space|<math>n</math>-connected]]. Another related notion is <em>[[Locally connected space|locally connected]]</em>, which neither implies nor follows from connectedness.
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