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• Calculus III -- Fall 2017
  *[http://users.marshall.edu/~saveliev/Teaching/Fall17/math231/set01.pdf Set 1] *[http://users.marshall.edu/~saveliev/Teaching/Fall17/math231/set2.pdf Set 2]
  10 KB (1,596 words) - 13:34, 27 November 2017
• Linear operators: part 4
  *$U={\bf R}^N$ with basis being the set of all vectors and *$W={\bf R}^n$ with basis being the set of all edges.
  13 KB (2,067 words) - 01:11, 12 September 2011
• Nerve of cover
  The open sets have non-empty pairwise intersections but the intersection of all three is empty. So what? Observe: *$n$-simplices corresponding to non-empty intersection of $n$ elements of the cover.
  8 KB (1,389 words) - 13:35, 12 August 2015
• Vector spaces: introduction
  Because it's not independent from the rest (and we want to make the set of axioms as short as possible). Indeed we derive it: '''Definition:''' A ''vector space'' is a set $V$, where two operations are defined:
  14 KB (2,238 words) - 17:38, 5 September 2011
• More on vector spaces
  Let $V$ be the set of all functions $f \colon {\bf R} \rightarrow {\bf R}$. But, we have to ask: is the set [[closed under operations|closed]] under these operations?
  14 KB (2,471 words) - 21:48, 5 September 2011
• Vector calculus: midterm
  ##The intersection of two linear subspaces is a linear subspace. ##The empty set is a linear subspace.
  4 KB (674 words) - 02:48, 22 August 2011
• Is the intersection of any collection of open sets always open?
  Q: Is the [[intersection]] of any collection of [[open set]]s always open? Under what circumstances ''is'' it open? [[intersection of a finite collection of open sets is open|Answer]].
  306 bytes (45 words) - 09:24, 3 September 2011
• Jordan theorem
  ...rdan theorem.''' ''The [[complement]] of a [[closed curve]] with no self-[[intersection]]s ("simple curve") in the [[plane]] has two [[connected components]].'' ...ion]] of a cubical complex in the plane is a [[open and closed sets|closed set]], hence its [[complement]] is open, so it's not a cubical complex.
  1 KB (200 words) - 09:26, 3 September 2011
• 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
• More metric complexes
  Furthermore, we connect the centers of the adjacent rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$. ...closed under the operations of complement, countable union, and countable intersection. If, furthermore, $K$ has a volume function, the function $\mu :\Sigma \to
  21 KB (3,445 words) - 13:53, 19 February 2016
• Geometric complexes
  Furthermore, we connect the centers of the adjacent rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$. ...closed under the operations of complement, countable union, and countable intersection. If, furthermore, $K$ has a volume function, the function $\mu :\Sigma \to
  20 KB (3,354 words) - 17:37, 30 November 2015
• 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
• 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
• 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
• 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
• 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

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