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• Connected sets
  26 bytes (2 words) - 20:59, 3 May 2011
• The union of any collection of open sets is open
  #REDIRECT [[Union of any collection of open sets is open]]
  58 bytes (10 words) - 13:58, 31 October 2010
• Disjoint sets
  Those are sets that don't intersect, i.e. have no elements in common. [[category:sets]]
  88 bytes (15 words) - 14:36, 9 October 2010
• Union of a finite collection of closed sets is closed
  ...collection of open sets is open]]. Closed sets are [[complement]] of open sets after all. It's a part of the definition of [[topological space]] in terms of closed sets, by the way.
  364 bytes (60 words) - 13:55, 31 October 2010
• Union of any collection of open sets is open
  445 bytes (73 words) - 13:58, 31 October 2010
• Upper and lower level sets
  #REDIRECT[[Lower and upper level sets]]
  39 bytes (6 words) - 22:13, 15 August 2009
• New sets from old
  [[Products of sets]]: Given two sets X and Y.
  977 bytes (182 words) - 15:01, 25 March 2010
• Intersection of a finite collection of open sets is open
  506 bytes (79 words) - 14:43, 23 February 2011
• Open and closed sets
  ...ds and topologies|neighborhoods]] $\gamma$ in $X$ is given. We define open sets as ones where every point has its own neighborhood: Some sets are neither closed nor open (unlike doors). Examples in in ${\bf R}$:
  4 KB (625 words) - 01:55, 1 October 2013
• Intersection of any collection of closed sets is closed
  ...collection of open sets is open]]. Closed sets are [[complement]]s of open sets after all. It's a part of the definition of [[topological space]] in terms of closed sets, by the way.
  359 bytes (60 words) - 13:58, 31 October 2010
• Open sets
  #REDIRECt[[Open and closed sets]]
  33 bytes (5 words) - 02:04, 14 October 2012
• Is the intersection of any collection of open sets always open?
  ...ircumstances ''is'' it open? [[intersection of a finite collection of open sets is open|Answer]].
  306 bytes (45 words) - 09:24, 3 September 2011
• Quotient sets
  [[Category:Sets]]
  3 KB (464 words) - 19:36, 31 October 2012
• Is the union of any collection of closed sets always closed?
  ...ances ''is'' such a union closed? [[union of a finite collection of closed sets is closed|Answer]].
  362 bytes (57 words) - 09:25, 3 September 2011
• Quotients of sets
  #redirect[[Quotient sets]]
  26 bytes (3 words) - 14:58, 15 February 2013
• Lower and upper level sets
  Or, these two sets for each T: They are also called "excursion sets" (why?).
  654 bytes (110 words) - 20:03, 6 September 2010
• Supralevel sets
  #redirect[[Lower and upper level sets]]
  39 bytes (6 words) - 19:24, 7 March 2011
• Compact sets
  24 bytes (2 words) - 20:59, 3 May 2011
• The union of a finite collection of closed sets is closed
  #REDIRECT [[Union of a finite collection of closed sets is closed]]
  67 bytes (11 words) - 13:55, 31 October 2010

Page text matches

• Functions
  ==Sets and relations== '''Example (lists).''' Sets given explicitly -- as lists -- are simplest ones:
  151 KB (25,679 words) - 17:09, 20 February 2019
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  ...ion]], respectively. The proposed method represents the hierarchy of these sets, and the [[topology]] of the image, by means of a graph. This graph contain ...The rationale for this approach is that the connected components of these sets are arguably building blocks of real items depicted in the image.
  41 KB (6,854 words) - 15:05, 28 October 2011
• Topological spaces
  ==Open and closed sets== and other related issues, all we need is to equip each of the sets involved with an additional structure called ''topology''.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Operations on functions
  ==Operations with sets== We can form a new set that contains all the elements of the two sets.
  142 KB (23,566 words) - 02:01, 23 February 2019
• The topology of a gray scale image
  ...ge. The boundaries of the objects are the level curves. Since all of these sets are connected collections of pixels, they will be represented as $0$- and $ Observe that all thresholds correspond to [[upper and lower level sets]].
  15 KB (2,589 words) - 12:31, 11 September 2013
• Neighborhoods and topologies
  ...ion into a basis, we'd have to add all ''singletons'', i.e., the one-point sets, to the collection. Since those are simply balls of diameter $0$, we end up The sets of all:
  16 KB (2,758 words) - 00:19, 25 November 2015
• Relative topology
  ...he topology<!--\index{topology}--> is given by a collection $\tau$ of open sets? Then how do we set up a topology for a subset? ...dex{open sets}--> in the $x$-axis $A$ can be seen as intersections of open sets in the $xy$-plane $X$:
  34 KB (6,089 words) - 03:50, 25 November 2015
• Tree representation of images
  ...]. A similar data structure is created for the holes (with the upper level sets). Combined these two graphs represent the topology of the image, the [[topo The [[connected components]] of [[upper and lower level sets]] of the [[gray scale function]] are building blocks of [[image segmentatio
  8 KB (1,263 words) - 18:45, 9 February 2011
• Manifolds
  A basis determines what sets are open in $X$. ..., we ''separate'' the two points from each other by means of disjoint open sets:
  51 KB (8,919 words) - 01:58, 30 November 2015
• Bases of neighborhoods
  ...ion into a basis, we'd have to add all ''singletons'', i.e., the one-point sets, to the collection. Since those are simply balls of diameter $0$, we end up The sets of
  11 KB (2,025 words) - 14:57, 2 August 2014
• A graph representation of the topology of color images by Saveliev
  ...vel function]] of the image. The rationale for this approach is that these sets are arguably building blocks of real items depicted in the image. Now we wo ...level sets may be objects and the connected components of the lower level sets may be holes in these objects. In addition to these, in order to capture th
  6 KB (1,011 words) - 15:33, 28 October 2011
• Open and closed sets
  ...ds and topologies|neighborhoods]] $\gamma$ in $X$ is given. We define open sets as ones where every point has its own neighborhood: Some sets are neither closed nor open (unlike doors). Examples in in ${\bf R}$:
  4 KB (625 words) - 01:55, 1 October 2013
• Continuous functions
  ...that we have developed, the '''definition''' becomes as follows. Given two sets $X,Y$ with bases of neighborhoods $\gamma_X,\gamma_Y$, a function $f:X\to Y We start with a simple observation that if we replace neighborhoods with open sets in the definition, it would still guarantee that the function is continuous
  42 KB (7,138 words) - 19:08, 28 November 2015
• Graph representation of gray scale images
  ...es [[adjacency]]). The [[connected components]] of [[upper and lower level sets]] of the [[gray scale function]] are building blocks of [[image segmentatio ...f a real object depicted in the image cutting through upper or lower level sets. However, one can imagine a picture with bald spot merging with the sky beh
  4 KB (653 words) - 04:45, 11 February 2011
• Triangulations
  ...hat if we aren't interested in these “small” open sets but in “large” open sets? We choose the latter to be unions of the interiors of simplices: Let $\gamma := \{U,V,W\}$ be this open cover of $X$. These sets came from the stars of the three vertices of $K$: ${\rm St}_A, {\rm St}_A,
  30 KB (5,172 words) - 21:52, 26 November 2015
• Quotients
  ==Quotient sets== ...re we consider the topological issues, let's take care of the underlying ''sets''<!--\index{quotient set}-->.
  26 KB (4,538 words) - 23:15, 26 November 2015
• Web search traffic
  <td class="TableCell">intersection of closed sets is closed proof</td> <td class="TableCell">the complement of a collection of closed sets is open</td>
  24 KB (3,456 words) - 13:01, 30 September 2011
• Products
  ...to every point on the $x$-axis. Or, we can think in terms of ''products of sets'': Generally, for any two sets $X$ and $Y$, their product set is defined as the set of ordered pairs taken
  44 KB (7,951 words) - 02:21, 30 November 2015
• Introduction to abstract mathematics: course
  *Sets: *New sets from old:
  3 KB (373 words) - 16:06, 25 September 2013
• Nerve of cover
  ...ntation if all you have is a topological space, i.e., a collection of open sets. ...sted in "small" open sets, i.e., ones inside simplices but in "large" open sets that are unions of the interiors of simplices.
  8 KB (1,389 words) - 13:35, 12 August 2015

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