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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]].
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• 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
• Real analysis: test 1
  #Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove. #Prove that an open ball in a metric space is an open set.
  1 KB (247 words) - 20:29, 13 June 2011
• Real analysis: test 3
  #Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove. #Prove that an open ball in a metric space is an open set.
  1 KB (247 words) - 20:30, 13 June 2011
• Introduction to Topology: Pure and Applied by Adams and Franzosa
  ** 0.4. Euclidean Space * 5. [[Metric Spaces]]
  3 KB (311 words) - 13:36, 26 October 2012
• Diffusion
  ...tment coefficient $\sin \widehat{ab}$. Note that when the metric tensor is Euclidean, $a$ and $b$ are perpendicular; then the coefficient is $1$. It is insufficient to have the two metric tensors, for a $K$ and $K^\star$:
  39 KB (6,850 words) - 15:29, 17 July 2015
• Test
  ...our study. The main topic is continuity, in the setting of $n$-dimensional Euclidean spaces. #[[Metric tensor]]
  16 KB (2,139 words) - 23:01, 9 February 2015
• Homeomorphism
  ...ndex{ isometry}-->, i.e., a bijection between two metric spaces<!--\index{ metric space}--> that preserves the distance, is a homeomorphism. ...that isn't a homeomorphism''. The reason is that our intuition takes us to Euclidean topology. Instead, consider the identity function
  13 KB (2,168 words) - 13:09, 7 August 2014
• Applied Topology and Geometry
  ...our study. The main topic is continuity, in the setting of $n$-dimensional Euclidean spaces. #[[Metric tensor]]
  16 KB (2,088 words) - 16:37, 29 November 2014
• Integer-valued calculus
  Invoking linear algebra now, we have Euclidean spaces with these generators and linear operators with these matrices: Invoking linear algebra now, we have Euclidean spaces with these generators and linear operators $\partial^*$ with these m
  27 KB (3,824 words) - 19:07, 26 January 2019
• Compact spaces
  ...ver $\alpha$ satisfying the above properties. Since the topology of $X$ is Euclidean, we can assume that the elements of $\alpha$ are the intersections of open '''Exercise.''' Prove the corollary for (a) metric spaces, and (b) all topological spaces. Hint: think of the sequence as a su
  19 KB (3,207 words) - 13:06, 29 November 2015
• Bases of neighborhoods
  ...er algebraic (such a [[vector space]]) or geometric structure (such as a [[metric space]]). In the $1$-dimensional Euclidean case, our neighborhoods $\gamma$ have been simply open intervals which make
  11 KB (2,025 words) - 14:57, 2 August 2014
• Point-set topology: course
  ...topologies, separation axioms, connectedness, compactness, continuity, and metric spaces. It is intended for advanced undergraduate and beginning graduate st *The topology of the [[Euclidean space]]
  2 KB (200 words) - 13:33, 17 March 2014
• Advanced Calculus II -- Spring 2017 -- final exam
  ...is the standard Euclidean metric of ${\bf R}^2$ while $d_1$ is the taxicab metric:
  1 KB (227 words) - 22:20, 4 May 2017
• Simplicial complexes and maps
  ...n order to see it, we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise ...an example of a “realization” of a $2$-dimensional simplicial complex as a metric complex in ${\bf R}^3$:
  27 KB (4,625 words) - 12:52, 30 March 2016
• Neighborhoods and topologies
  ...ave to have either algebraic (such a vector space) or geometric (such as a metric space) structure. In the case of the $1$-dimensional Euclidean space ${\bf R}$, our neighborhoods $\gamma$ have been simply open intervals
  16 KB (2,758 words) - 00:19, 25 November 2015
• Homology as an equivalence relation
  <center>''subsets of a Euclidean space'', $X\subset {\bf R}^N$.</center> [[image:metric complex 1d w worm.png|center]]
  21 KB (3,530 words) - 19:54, 23 June 2015
• Homology classes
  <center>''subsets of a Euclidean space'', $X\subset {\bf R}^N$.</center> <!--150-->[[image:metric complex 1d w worm.png|center]]
  21 KB (3,581 words) - 15:51, 28 November 2015
• Topology 1: course
  ...topologies, separation axioms, connectedness, compactness, continuity, and metric spaces. It is intended for advanced undergraduate and beginning graduate st *The topology of the [[Euclidean space]]
  2 KB (204 words) - 13:33, 17 March 2014
• Topology I -- Fall 2013
  ...0 - Topology I.''' First course in topology. Basics of point-set topology: metric and topological spaces, continuity, connectedness, compactness, products, q #[[Discretization of the Euclidean space]]
  7 KB (881 words) - 19:04, 10 December 2013
• Triangle inequality
  In [[Euclidean space]]s: In [[metric space]]s:
  258 bytes (41 words) - 23:34, 14 February 2011
• Isotropy in numerical PDEs
  ...t surprising as this square ''is'' a circle, with respect to the [[taxicab metric]].) Therefore, $\pi=4$! (the circumference of the circle divided by twice t We check the Euclidean theorem:
  10 KB (1,593 words) - 13:20, 8 April 2013
• Links
  *[[Forms in Euclidean spaces]] It is insufficient to have the two metric tensors, for a $K$ and $K^\star$:
  6 KB (998 words) - 12:40, 31 August 2015
• Circle
  In [[Euclidean geometry]]: In [[metric space]] $(M,d)$:
  797 bytes (127 words) - 14:26, 22 February 2011
• Convergence of the discrete to the continuous
  [[image:digital Euclidean curve.png| center|Length of a digital curve.]] *$\gamma:=\{K\}$ is the set of all metric cell complexes each representing ${\bf R}^n$.
  21 KB (3,664 words) - 02:02, 18 July 2018
• Set-valued maps
  ...d prove the analogue of the proposition where ${\bf R}$ is replaced with a metric space. ...int Theorem).''' Let $X$ be a non-empty, compact and convex subset of some Euclidean space. Let $F: X \to X$ be a set-valued map with a polyhedral graph and eac
  24 KB (4,382 words) - 15:52, 30 November 2015