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• Functions
  ...ppose there is ''another'', unrelated, set, say $Y$, the set of these four balls: [[image:four balls.png| center]]
  151 KB (25,679 words) - 17:09, 20 February 2019
• 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
• Relative topology
  *the Euclidean basis of $X:={\bf R}^2$ is the open disks; *the Euclidean basis of $A:={\bf R}$ is the open intervals.
  34 KB (6,089 words) - 03:50, 25 November 2015
• Neighborhoods and topologies
  ...nal Euclidean space ${\bf R}$, our neighborhoods $\gamma$ have been simply open intervals, which makes the refining condition of $\gamma$ especially short: Now, we can modify this collection $\gamma$ by choosing open intervals with the end-points:
  16 KB (2,758 words) - 00:19, 25 November 2015
• Bases of neighborhoods
  ...1$-dimensional Euclidean case, our neighborhoods $\gamma$ have been simply open intervals which makes the refining condition of $\gamma$ is especially simp We can modify this $\gamma$ by choosing open intervals with only rational end-points, or irrational, etc. However, close
  11 KB (2,025 words) - 14:57, 2 August 2014
• Continuous functions
  where $B(p,d)= \{u :\ ||u-p|| < d \}$ is an open ball in ${\bf R}^n$: As the last step, we interpreted these Euclidean balls as “neighborhoods” of points, i.e., elements of a basis that generates
  42 KB (7,138 words) - 19:08, 28 November 2015
• Manifolds
  ...--\index{locally Euclidean space}--> $n$ if for every $x\in X$ there is an open set $U$ such that $x\in U$ and there is a homeomorphism $h:{\bf R}^n \to U$ ...se, “homeomorphic to ${\bf R}^n$” can be replaced with “homeomorphic to an open $n$-ball”, or “box”, etc.:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Topological spaces
  ==Open and closed sets== ...gical space}-->. The elements of $\tau$ are called ''open sets''<!--\index{open sets}-->.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Open and closed sets
  ...orhoods and topologies|neighborhoods]] $\gamma$ in $X$ is given. We define open sets as ones where every point has its own neighborhood: [[Image:open set definition.png|500px|center]]
  4 KB (625 words) - 01:55, 1 October 2013
• 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
  These sets are open intervals: ...1$-dimensional “balls”<!--\index{balls}-->, while in general, we define an open ball in ${\bf R}^n$ as:
  17 KB (2,946 words) - 04:51, 25 November 2015
• Topology Illustrated -- Index
  *[[balls|balls]] *[[closed balls|closed balls]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• 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
• Vector fields
  It is as if we cover the whole stream with those little balls and study their rotation. [[image:Cube with balls.png| center]]
  91 KB (16,253 words) - 04:52, 9 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
• Products
  $$\{U \times V : U \text{ open in } X,\ V \text{open in } Y \}.$$ ...to work, we need to show that compactness holds even if we only deal with open covers of a particular kind.
  44 KB (7,951 words) - 02:21, 30 November 2015
• From continuity to point-set topology
  These sets are open intervals but can be also seen as "balls": *$\gamma _X$ is the set of all open balls in $X={\bf R}^n$, and
  7 KB (1,207 words) - 13:01, 12 August 2015
• Functions of several variables
  ...'parabola''. And so is the part that lies above the line $y=-x$; it's just open down instead of up: ...the limit at $X=A$ of function $z=f(X)$ exists then $f$ is bounded on some open disk that contains $A$:
  97 KB (17,654 words) - 13:59, 24 November 2018
• Real analysis: final 2
  #Prove that an open ball in a metric space is an open set. ...)$ in Problem 2 are Euclidean, $S_{1}=S_{2}=\mathbf{R.}$ Describe the open balls in $(T,D),$ convergent sequences, completeness, compactness, and connectedn
  4 KB (582 words) - 20:29, 13 June 2011
• Euclidean space made discrete
  <!--s-->[[image:balls pixelated.png|center]] ...aren't closed! To avoid these abnormalities, it makes sense to consider ''open'' subsets of ${\bf R}^n$. The advantage is that every point has a neighborh
  11 KB (1,801 words) - 15:50, 25 July 2014

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