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• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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

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