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• Metric complexes
  '''Proposition.''' A normed space is a metric space with the metric $d(x,y):=\lVert x-y \rVert$. '''Exercise.''' Prove that the inner product is continuous in this metric space.
  35 KB (5,871 words) - 22:43, 7 April 2016
• Topological spaces
  “Open” disks on the plane, and balls in the Euclidean space, are also open. “Closed” disks on the plane, and “closed” balls in the Euclidean space are also closed. Points too.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Metric tensor
  '''Proposition.''' A normed space is a metric space with the metric $d(x,y):=\lVert x-y \rVert.$ '''Exercise.''' Prove that the inner product is continuous in this metric space.
  42 KB (7,131 words) - 17:31, 30 November 2015
• Functions in higher dimensions
  ==Euclidean spaces and Cartesian systems of dimensions $1$, $2$, $3$,...== '''Theorem (Axioms of Metric Space).''' Suppose $P,Q,S$ are points in ${\bf R}^3$. Then the following pr
  113 KB (19,680 words) - 00:08, 23 February 2019
• Manifolds
  The space of locations of the Newtonian physics is the $3$-dimensional Euclidean space, ${\bf R}^3$. We understand the topology<!--\index{topology}--> of th These “locally Euclidean”<!--\index{locally Euclidean}--> spaces are called ''manifolds''<!--\index{manifold}-->.
  51 KB (8,919 words) - 01:58, 30 November 2015
• More metric complexes
  Let's recall the mechanical interpretation of a realization $|K|$ of a metric complex $K$ of dimension $n=1$: ==Metric complexes==
  21 KB (3,445 words) - 13:53, 19 February 2016
• Topology Illustrated -- Index
  *[[Euclidean metric|Euclidean metric]] *[[Euclidean topology|Euclidean topology]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Relative topology
  ...d simply by the distance formula, the Euclidean metric<!--\index{Euclidean metric}-->. The distance between $(x,y)$ and $(a,b)$ is which is the Euclidean metric of the $x$-axis. Not by coincidence, the proximity of points in $A$ is meas
  34 KB (6,089 words) - 03:50, 25 November 2015
• Calculus exercises: advanced
  ...is the standard Euclidean metric of ${\bf R}^2$ while $d_1$ is the taxicab metric: $$d_1\left( (x,y),(u,v) \right)=|u-x|+|v-y|.$$ Do one of the two: (a) prov * Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.
  14 KB (2,538 words) - 18:35, 14 October 2017
• Real analysis: final 2
  #Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove. #Prove that an open ball in a metric space is an open set.
  4 KB (582 words) - 20:29, 13 June 2011
• Geometric complexes
  <!--350-->[[image:metric tensor vs dual complex.png|center]] ...dimension $n=1$ as cell complexes equipped with a geometric structure: the metric tensor of $K$ and $K^\star$. This data allows one to compute the lengths an
  20 KB (3,354 words) - 17:37, 30 November 2015
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...n the [[metric]] acquired from the [[embedding]] of the [[manifold]] n the Euclidean space. The [[topological data analysis|topological approach]] would be to u ...system is made up. And there are [[Non-Euclidean topology on the plane|non-Euclidean topologies]] too.
  11 KB (1,663 words) - 16:03, 26 November 2012
• Geometry of data
  ...data is finding [[k-nearest neighbor]]s of each node and then using the [[Euclidean distance]]s. ...on the plane|non-Euclidean topologies]] for the Euclidean space, even with Euclidean topology on each of the coordinates.
  984 bytes (149 words) - 22:27, 26 February 2011
• Real analysis: final 1
  #Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove. #Prove that an open ball in a metric space is an open set.
  3 KB (562 words) - 20:29, 13 June 2011
• Metric space
  ...real valued function $d: X \times X \longrightarrow {\bf R}$ (called a ''[[metric]]'') such that, for every $x,y,z \in X$, ...$X$ generated by these sets (as a [[basis of topology]]) is called the ''[[metric topology]]''. This topology is [[Hausdorff]].
  819 bytes (155 words) - 05:11, 18 February 2011
• Topological data analysis
  ...sconnected 1000 points, called a [[point cloud]], in the 100-dimensional [[Euclidean space]]. For a point cloud in a [[euclidean space]], suppose we are given a threshold $r$ so that any two points within
  4 KB (549 words) - 12:54, 12 August 2015
• Continuous functions
  As the last step, we interpreted these Euclidean balls as “neighborhoods” of points, i.e., elements of a basis that gene '''Proposition.''' In case of metric spaces, a function $f:X\to Y$ is continuous if and only if ''the function c
  42 KB (7,138 words) - 19:08, 28 November 2015
• Point-set topology: exercises
  ...ntinuous functions $f:[0,1]\rightarrow {\bf R}$ is a metric space with the metric $d(f,g)=\max |f(x)-g(x)|$. #Is $D(E,E')=\max \{\min \{d(x,y),1\}:x \in E, y \in E'\}$ a metric on the quotient space?
  5 KB (814 words) - 16:40, 4 October 2013
• Metric
  ...et $X$, any function $d: X \times X \longrightarrow {\bf R}$ is called a ''metric'' (or a "distance function") if, for every $x,y,z \in X$, ...alled the ''[[triangle inequality]]''.) In this case $(X,d)$ is called a [[metric space]].
  531 bytes (104 words) - 05:14, 18 February 2011
• Products
  ...f R}$ is based on intervals. The solution was demonstrated previously: the Euclidean topology of the plane coincides with the topology generated by ''rectangles '''Exercise.''' Prove that the metric $d$ of a metric space $(X,d)$ is continuous as a function $d:X\times X\to {\bf R}$ on the p
  44 KB (7,951 words) - 02:21, 30 November 2015

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