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• Constructions
  We use the following '''notation''' for the quotient set: As we know, it is simply the set of all equivalence classes of this equivalence relation:
  13 KB (2,270 words) - 22:14, 18 February 2016
• Quotient sets
  i.e., the set of equivalence classes: ...relation $~$ on it, the corresponding ''quotient set'' $X/_{\sim}$ is the set of its equivalence classes:
  3 KB (464 words) - 19:36, 31 October 2012
• Product topology
  ...lead, for any two sets $X$ and $Y$ their [[product set]] is defined as the set of ordered pairs taken from $X$ and $Y$: ...X \times Y$ of $X$ and $Y$ is a topological space defined on the [[product set]] $X \times Y$ with the following [[Neighborhoods and topologies|basis]]:
  8 KB (1,339 words) - 16:53, 27 August 2015
• Introduction to abstract mathematics: course
  **the [[power set]], **[[union]]s, [[intersection]]s, [[complement]]s
  3 KB (373 words) - 16:06, 25 September 2013
• Linear Algebra -- Spring 2013 -- final exam
  ...the main diagonal are equal to zero ($a_{ij}=0$ for $i>j$). Prove that the set of all upper-triangular $n\times n$ matrices form a vector space. #Prove that the intersection of two subspaces is always a subspace.
  2 KB (330 words) - 02:21, 7 May 2013
• Manifolds model a curved universe
  ...''' If every point $a \in C$ has an [[open and closed sets|open]] (in $C$) set $U$, $a \in U$, [[homeomorphic]] to ${\bf R}^1$, then $C$ is called a ''$1$ ...$S$ is a subset of ${\bf R}^N$ such that every point $a \in S$ has an open set $U$ such that $a \in U$ and $U$ is homeomorphic to ${\bf R}^2$.
  10 KB (1,588 words) - 17:11, 27 August 2015
• Examples of maps
  ...nctions|continuous]]. The proof of that below relies on the relevant point-set topology. However, it's unnecessary if only subsets of the Euclidean space As the intersection an open, in $X$, set with $A$, this set is open in $A$. Hence $f_A$ is continuous. $\blacksquare$
  5 KB (918 words) - 16:54, 27 August 2015
• Homeomorphism
  <center>given $x \in X$, draw a line through $x$ and $N$, find its intersection $y$ with $Y$, then $y = f(x)$.</center> ...ates an [[equivalence relation]]<!--\index{equivalence relation}--> on the set of all [[topological space]]s<!--\index{ topological space}-->.
  13 KB (2,168 words) - 13:09, 7 August 2014
• Homology of cubical complexes
  '''Definition.''' The $k$-th ''homology'' $H_k(K)$ of complex $K$ is the set of all $k$-homology classes of $K$. ...cells as parking lots, $0$-homology as a turn-by-turn instructions from an intersection to another. Solution: [[Image:0-hom and path-con.jpg SOLUTION.jpg|15px]]
  7 KB (1,118 words) - 12:58, 12 August 2015
• Calculus exercises: part II
  *Set up the Riemann sum for the area of the circle of radius $R$ as the area bet *Suppose the parametric curve is given by \[x=\cos3t,\ y=2\sin t.\] Set up, but do not evaluate, the integrals that represent (a) the arc-length of
  15 KB (2,591 words) - 17:15, 8 March 2018
• Parametric curves
  '''Problem.''' Give the set of all [[affine function]]s passing through $(a,f(a))$ '''Definition.''' Given a set of real numbers $S$, an ''upper bound'' $n$ of $S$ is a number $n < \infty$
  34 KB (5,665 words) - 15:12, 13 November 2012
• Vector calculus: review
  #Give an example of a set $S$ and a point $p\in S$ such that $p$ is a limit point of $S$ and but not ##The intersection of two linear subspaces is a linear subspace.
  7 KB (1,394 words) - 02:36, 22 August 2011
• Transformations
  ...lines on the plane. Then the solution $(x,y)=(4,2)$ is the point of their intersection: ...he ''integrand'' (and the output is another parametric curve) but a way to set up the ''domain of integration'' (and the output is a number).
  46 KB (7,625 words) - 13:08, 26 February 2018
• Union of a finite collection of closed sets is closed
  Prove that the [[union]] of a finite collection of [[closed set]]s is closed. It follows from the fact that [[intersection of a finite collection of open sets is open]]. Closed sets are [[complement
  364 bytes (60 words) - 13:55, 31 October 2010
• Intersection of any collection of closed sets is closed
  Prove that the [[intersection]] of any collection of [[closed set]]s is closed.
  359 bytes (60 words) - 13:58, 31 October 2010
• Homology and algebra
  '''Definition.''' The $k$-th ''homology'' $H_k(K)$ of complex $K$ is the set of all $k$-homology classes of $K$. ...cells as parking lots, $0$-homology as a turn-by-turn instructions from an intersection to another. Solution: [[Image:0-hom and path-con.jpg SOLUTION.jpg|15px]]
  8 KB (1,386 words) - 18:40, 27 August 2015
• Computational Topology by Edelsbrunner and Harer
  ** [[Convex]] set systems 57 ** [[Intersection theory]] 114
  1 KB (101 words) - 20:23, 5 November 2012
• Basis of topology
  Suppose a set $X$ is given. Any collection $\gamma$ of subsets of $X$ is called a ''basis [[Image:intersection of nbhds.jpg|right]]
  816 bytes (143 words) - 18:41, 2 October 2013

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