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• Vector fields
  ==Sums along closed curves reveal exactness== What can we say about the sums of such functions along ''closed'' curves?
  91 KB (16,253 words) - 04:52, 9 January 2019
• Manifolds
  ...sider these examples of dimension $1$ locally Euclidean in space. They are closed, open, and half-open; finite and infinite; and the disjoint unions of these ...the standard “local” basis at $a$, which is the set of all open $\epsilon$-balls around $a$:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Topological spaces
  ==Open and closed sets== “Open” disks on the plane, and balls in the Euclidean space, are also open.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Combinatorial cell complexes
  ...f things created from elementary pieces appear in real life, such as these balls sewn (or otherwise formed) from patches of leather: <!--150-->[[image:balls as cell complexes.png| center]]
  34 KB (5,710 words) - 22:27, 18 February 2016
• A new look at continuity
  We look at these intervals as $1$-dimensional “balls”<!--\index{balls}-->, while in general, we define an open ball in ${\bf R}^n$ as: ...For a function of two variables, what if the domain is a closed square, a closed disk?
  17 KB (2,946 words) - 04:51, 25 November 2015
• Relative topology
  ...in difference is that for vector spaces (or groups) the subset needs to be closed under the operations while for topological spaces a subset can ''always'' b ...d, all the topological concepts become available. We can speak of open and closed sets “in $A$” (as opposed to “in $X$”), the interior, exterior, clo
  34 KB (6,089 words) - 03:50, 25 November 2015
• Topology Illustrated -- Index
  *[[balls|balls]] *[[closed balls|closed balls]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Triangulations
  ...are homeomorphic to points, closed segments, disks, balls, ..., closed $n$-balls. ...al) complexes are ''closed'', i.e., homeomorphic to closed balls<!--\index{balls}-->. The reason is that a cubical complex may be built as the union of a co
  30 KB (5,172 words) - 21:52, 26 November 2015
• Cell complex
  ...f things created from elementary pieces appear in real life, such as these balls sewn from patches of leather: <!--150-->[[image:balls as cell complexes.png|center]]
  40 KB (6,459 words) - 23:27, 29 November 2015
• Open and closed sets
  "Open" disks on the plane, and balls in the [[Euclidean space]] are also open. '''Definition.''' A set is called ''closed'' if its [[complement]] is open.
  4 KB (625 words) - 01:55, 1 October 2013
• Neighborhoods and topologies
  However, closed intervals (of non-zero length) won't work: The collection of all ''open balls''<!--\index{open balls}--> in $X = {\bf R}^n$:
  16 KB (2,758 words) - 00:19, 25 November 2015
• Integer-valued calculus
  On the other hand, the line integral over a closed path of a gradient vector field would be $0$ according to the ''Fundamental On the other hand, the line integral over a closed path of a gradient vector field would be $0$ according to the ''Fundamental
  27 KB (3,824 words) - 19:07, 26 January 2019
• Classes of functions
  *$Y$ is the four balls. [[image:boys and balls -- constant.png| center]]
  143 KB (24,052 words) - 13:11, 23 February 2019
• Continuous functions
  As the last step, we interpreted these Euclidean balls as “neighborhoods” of points, i.e., elements of a basis that generates *closed domain, such as $[a,b]$.
  42 KB (7,138 words) - 19:08, 28 November 2015
• Chain complexes
  A subgroup $N$ of $M$ is a ''submodule'' if it is closed under scalar multiplication: for any $n \in N$ and any $r\in R$, we have $r ...uppose also that we have its decomposition ${\mathbb R}^N$, a list of all (closed) cubical cells in our grid.
  33 KB (5,293 words) - 03:06, 31 March 2016
• Bases of neighborhoods
  ...open intervals with only rational end-points, or irrational, etc. However, closed intervals (of non-zero length) won't work as The collection of all ''open balls'' in $X = {\bf R}^n$:
  11 KB (2,025 words) - 14:57, 2 August 2014
• Euclidean space made discrete
  <!--s-->[[image:balls pixelated.png|center]] '''Exercise.''' Show that the [[Jordan theorem]]: the complement of a closed curve with no self-intersections in the plane has two connected components,
  11 KB (1,801 words) - 15:50, 25 July 2014
• Real analysis: final 2
  #Prove that a compact set in a metric space is bounded and closed. ...$ are continuous functions. Prove that the set $A=\{x\in S:f(x)=g(x)\}$ is closed in $S.$ What can you say about $B=\{x\in S:f(x)\neq g(x)\}?$
  4 KB (582 words) - 20:29, 13 June 2011
• Functions of several variables
  But what is the multi-dimensional analog of a closed bounded interval? and a set in ${\bf R}^n$ is called ''closed'' if it contains the limits of all of its convergent sequences.
  97 KB (17,654 words) - 13:59, 24 November 2018
• Applied Topology and Geometry
  #[[Closed and exact forms]] #[[Open and closed sets]]
  16 KB (2,088 words) - 16:37, 29 November 2014

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