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• Manifolds
  ...ce, ${\bf R}^3$. We understand the topology<!--\index{topology}--> of this space quite well. Most important is that it's acyclic<!--\index{acyclic}-->. Howe ...ial homology groups<!--\index{homology groups}-->. Our conclusion is that, locally, the universe looks like ${\bf R}^3$, but possibly not globally.
  51 KB (8,919 words) - 01:58, 30 November 2015
• Compact spaces
  ...nd $f(b)$. It follows from this theorem that the image of a path-connected space<!--\index{path-connectedness}--> (under a continuous map<!--\index{continuo ...will rely on the following familiar concept. A point $x$ in a topological space $X$ is called an accumulation point<!--\index{accumulation point}--> of sub
  19 KB (3,207 words) - 13:06, 29 November 2015
• Topology Illustrated -- Index
  *[[acyclic space|acyclic space]] *[[compact space|compact space]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Forms in Euclidean spaces
  ...space of continuous $k$-forms is denoted by $\Omega^k({\bf R}^n)$ and the space of discrete forms is $T^k({\mathbb R}^n)$. The above argument applies to show that in $3$-space the direction variables are independent from the location variables $x$, $y
  44 KB (7,778 words) - 23:32, 24 April 2015
• Surfaces and their homology
  What do they have in common? Locally, they look like [[disk]]s. Just like the Earth looks like a plane if you do ...dimensional [[Euclidean space]] ${\bf R}^2$, we can say that surfaces are "locally Euclidean".
  17 KB (2,696 words) - 00:47, 12 January 2014
• Surface
  What do they have in common? Locally, they look like [[disk]]s. Just like the Earth looks like a plane if you do ...dimensional [[Euclidean space]] ${\bf R}^2$, we can say that surfaces are "locally Euclidean".
  5 KB (718 words) - 18:16, 27 August 2015
• Manifolds model a curved universe
  In other words, locally, the universe is like ${\bf R}^3$, but not globally. What's locally like ${\bf R}^1$ but not globally?
  10 KB (1,588 words) - 17:11, 27 August 2015
• Fixed points and selections of set valued maps on spaces with convexity by Saveliev
  ...ompact]] subset of a [[locally convex]] [[Hausdorff]] [[topological vector space]], and let $F:X \rightarrow Y$ be an [[upper semicontinuous]] [[multifuncti ...$X$ be a nonempty convex compact subset of a Hausdorff topological vector space, and let $F:X\rightarrow X$ be a multifunction with nonempty convex images
  3 KB (469 words) - 16:12, 26 March 2011
• Differential equations
  ...are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: ...placed in the first row of the spreadsheet and, as we progress in time and space, new numbers are placed in the next row of our spreadsheet:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Introduction to Topology by Gamelin and Greene
  Could use more pictures. Proofs can be more gentle, details, less "compact". 7 [[Locally compact space]]s
  2 KB (170 words) - 21:51, 4 May 2011
• More about manifolds
  #The manifold is [[compact]] if and only if it has a finite number of cells. *locally constant = 2 dimensional.
  9 KB (1,542 words) - 19:58, 21 January 2014