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• Functions
  ...to any loose collection of objects or entities -- of any nature -- as a ''set''. ...lly, whether the items are arranged in a box or piled up on the counter. A set of encyclopedia consists of books that can be arranged alphabetically or ch
  151 KB (25,679 words) - 17:09, 20 February 2019
• Operations on functions
  ...ack to our the example of five boys that form a set and another set is the set of these four balls: 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
• Topological spaces
  '''Definition.''' Given an arbitrary set $X$, a collection $\tau$ of subsets of $X$ is called a ''topology<!--\index Of course, instead, we can, and did, start with the set of all neighborhoods $\gamma$ in $X$ and then define open sets as ones wher
  27 KB (4,693 words) - 02:35, 20 June 2019
• Linear algebra
  '''Theorem 1.4.3''' The solution set of a homogeneous linear equation ...''a''₃ = 0, the the equation turn into 0 = 0, and the solution set is '''R³'''.
  27 KB (4,667 words) - 01:07, 19 February 2011
• Linear algebra of Euclidean space
  '''Theorem 1.4.3''' The solution set of a homogeneous linear equation If $a_1=a_2=a_3=0$, the the equation turns into $0 = 0$, and the solution set is ${\bf R}^3$.
  26 KB (3,993 words) - 19:48, 26 August 2011
• Neighborhoods and topologies
  '''Definition.''' Suppose $X$ is any set. A collection $\gamma$ of subsets of $X$ satisfying the properties of: ...r any neighborhoods $U, V\in \gamma$ and any point $x\in U\cap V$ in their intersection, there is a neighborhood $W\in \gamma$ of $x$ such that $W \subset U \cap V
  16 KB (2,758 words) - 00:19, 25 November 2015
• Relative topology
  ...s produce intervals and, conversely, every interval in the $x$-axis is its intersection with a disk: ...$X$ with basis $\gamma$ will acquire its own collection $\gamma _A$ as the set of all of its intersections with the elements of $\gamma$:
  34 KB (6,089 words) - 03:50, 25 November 2015
• Data made Euclidean
  This set is called a ''realization''<!--\index{realization}--> of the graph which is This data set is called a ''simplicial complex''<!--\index{simplicial complex}--> (or som
  31 KB (5,219 words) - 15:07, 2 April 2016
• Classes of functions
  ...s a ''constant function'', i.e., for some specified element $b$ of $Y$, we set: So, for our set $X$ of boys, we have a special function $G$ from $X$ to $X$ (and another fr
  143 KB (24,052 words) - 13:11, 23 February 2019
• Products
  Generally, for any two sets $X$ and $Y$, their product set is defined as the set of ordered pairs taken from $X$ and $Y$: ...s to demonstrate how we can define the algebraic operations on the product set ${\bf R} \times {\bf R}$ in terms of the operations on either copy of ${\bf
  44 KB (7,951 words) - 02:21, 30 November 2015
• Applications of integral calculus
  ...on of two intervals is the sum of the two lengths minus the lengths of the intersection: ...the union of two regions is the sum of the two areas minus the area of the intersection:
  103 KB (18,460 words) - 01:01, 13 February 2019
• Triangulations
  ...if we don't even have a cell complex but only a topological space, i.e., a set with a collection of open subsets? What topological spaces are polyhedra? ...pen sets have non-empty ''pairwise'' intersections but the only ''triple'' intersection is empty. Why is this important? Notice the pattern:
  30 KB (5,172 words) - 21:52, 26 November 2015
• Quotients
  ...ical issues, let's take care of the underlying ''sets''<!--\index{quotient set}-->. Now, '''notation''' for the quotient set is:
  26 KB (4,538 words) - 23:15, 26 November 2015
• Continuous functions
  This set is defined explicitly and is independent from the existence of the inverse ...appear implicitly in many areas of mathematics. For example, the solution set of the equation $f(x) = b$ is $f^{-1}(\{b\})$. In particular, the kernel of
  42 KB (7,138 words) - 19:08, 28 November 2015
• Bases of neighborhoods
  '''Definition.''' Suppose $X$ is any set. Any collection $\gamma$ of subsets of $X$ satisfying properties: ...r any neighborhoods $U, V\in \gamma$ and any point $x\in U\cap V$ in their intersection, there is a neighborhood $W\in \gamma$ of $x$ such that $W \subset U \cap V
  11 KB (2,025 words) - 14:57, 2 August 2014
• Open and closed sets
  Suppose the set of all [[Neighborhoods and topologies|neighborhoods]] $\gamma$ in $X$ is gi [[Image:open set definition.png|500px|center]]
  4 KB (625 words) - 01:55, 1 October 2013
• Manifolds
  ...dex{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 $h:{\bf R}^n \to U$. Th ...n we would simply consider 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
• Functions of several variables
  '''Definition.''' The ''graph'' of a function of one variable $z=f(x)$ is the set of all points in ${\bf R}^{2}$ of the form $(x,f(x))$. ...inition.''' The ''graph'' of a function of two variables $z=f(x,y)$ is the set of all points in ${\bf R}^{3}$ of the form $(x,y,f(x,y))$.
  97 KB (17,654 words) - 13:59, 24 November 2018
• Calculus exercises: part III
  '''Example.''' Suppose $A$ and $B$ are path-connected and $a$ is in the intersection. If $P$ and $Q$ belong to the union, find a path from $P$ to $a$, from $a$ *Set up, but do not solve, a system of linear equations for the following proble
  46 KB (8,035 words) - 13:50, 15 March 2018
• Differentiation
  We set up two functions, for the location and the altitude. Then their composition ...t_1$ your car is the same distance from the intersection as it is past the intersection at time $t_2$. Then the $\Delta S=0$! So, the reading ''can'' be off by a l
  82 KB (14,116 words) - 19:50, 6 December 2018

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