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• Compactness
  * Every [[open covering]] of {{M|X}}, {{M|\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J} } * Every [[covering]] by sets [[open set|open]] in {{M|X}} of {{M|S}} contains a ''finite'' [[sub-cover]]
  5 KB (828 words) - 15:59, 1 December 2015
• Index of notation
  ...le <math>C([a,b],\mathbb{R})</math> denotes the continuous function on the interval {{M|[a,b]}} that map to {{M|\mathbb{R} }} - this is unlikely to be given an | "Closed under", "Open in"
  9 KB (1,490 words) - 06:13, 1 January 2017
• Semi-ring of half-closed-half-open intervals
  ...length (namely {{M|n\in\mathbb{N} }}), we define {{M|1=\[a,b\)}}, a ''half-open-half-closed rectangle in {{M|\mathbb{R}^n}}''{{rMIAMRLS}} as follows: ...in\mathbb{R}\ \vert\ \alpha\le x < \beta\} }}<ref group="Convention">For [[interval|intervals]] in general we define the following:
  4 KB (680 words) - 00:23, 20 August 2016
• Notes:Just what is in a generated sigma-algebra
  ...[[Implies and subset relation|implies-subset relation]]) every [[Open set|open set]] is in {{M|\sigma(\mathcal{J})}}. We also know that {{M|\sigma(\mathca ...operations a [[Sigma-algebra|{{Sigma|algebra}}]] allows starting from the open sets, {{M|\mathcal{J} }}
  1 KB (256 words) - 13:29, 17 June 2015
• Topological property theorems
  ! {{M|V}}-Open ! {{M|V}}-Open
  3 KB (616 words) - 08:37, 1 July 2015
• Notes:Spherical coordinates
  ...k around to zero when you're by {{M|360}} - but this is an open end of the interval, which means for an {{M|x\in(0,360)}} - any {{M|x}} you like - there is a s
  10 KB (1,899 words) - 18:48, 23 September 2015
• Given a topological manifold of dimension 2 or more and points p1, p2 and q where q is neither p1 nor p2 then a path from p1 to p2 is path-homotopic to a path that doesn't go through q
  ...a path going through it, and another picture of the path but moved in the open ball to avoid {{M|q}}}} ...|f:I\rightarrow M}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[unit interval]]) from {{M|p_1}} to {{M|p_2}}, is [[path-homotopic]] to a path that ''does
  954 B (165 words) - 11:57, 10 May 2016
• Notes:Mean value theorem
  ====For {{M|f:[x,x+h]\subseteq U\rightarrow\mathbb{R} }} for {{M|U}} open in a [[Banach space]]==== ...|(X,\Vert\cdot\Vert)}}, if {{M|f}} is differentiable at every point of the interval {{M|[x,x+h]}} {{Note|For what {{M|h}}?}} then<ref name="APIKM"/>:
  3 KB (529 words) - 08:07, 4 June 2016
• Semi-ring of sets
  ...thscr{J}^1}}, here is the collection of all half-open-half-closed {{plural|interval|s}} on the [[real line]], {{M|[a,b)\subset\mathbb{R} }} (with {{M|1=[a,b):= ...{{M|a,b,c,d\in\mathbb{R} }}, Suppose {{M|a<b}} and {{M|c<d}} (as if either interval is the empty set the result is trivial). Suppose they partially intersect w
  3 KB (508 words) - 17:25, 18 August 2016
• Constant loop based at a point
  ...X}}<ref group="Note">Where {{M|1=I:=[0,1]\subset\mathbb{R} }} (the [[unit interval]])</ref> such that: #* Let {{M|U\in\mathcal{J} }} be given (so {{M|U}} is an [[open set]] in {{Top.|X|J}})
  3 KB (479 words) - 21:03, 1 November 2016
• Example:Smooth map between the real line and the circle (considered as smooth manifolds)
  * {{M|I:\eq(-1,1)\subset\mathbb{R} }} - the open interval of unit length in each direction<ref group="Note">typically we use {{M|I:\e ...ious"<ref group="Note">{{XXX|Qualify this!}}</ref> that restricted to an [[open subset]] of {{M|\mathbb{R} }} would leave smoothness at a point unhindered.
  4 KB (757 words) - 13:25, 2 April 2017
• Expectation of the geometric distribution
  ...that that {{M|\E{X}\eq\frac{1}{p} }} for {{M|p\in}}[[Half-open-half-closed interval|{{M|(0,1]}}]]{{M|\subseteq\mathbb{R} }} and undefined for {{M|p\eq 0}} * Let {{M|p\in}}[[open interval|{{M|(0,1)}}]]{{M|\subseteq}}[[Reals|{{M|\mathbb{R} }}]] be given, and let {
  8 KB (1,522 words) - 02:55, 16 January 2018