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• Main Page
  ...s. This means they are written verbosely and with plenty of explanation in order help those new to the topic. A good example is the [[Limit (sequence)|limit * [[Topology]]
  7 KB (999 words) - 16:51, 11 May 2020
• Topology
  ...ies we don't mean a topology but rather a [[topological space]] which is a topology with its underlying set. See that page for more details}} A ''topology'' on a [[set]] {{M|X}} is a collection of [[subset|subsets]], {{M|J\subsete
  3 KB (543 words) - 09:28, 30 December 2016
• Homeomorphism
  {{Refactor notice|grade=A|msg=As a part of the topology patrol. ...then say that {{M|X}} and {{M|Y}} (or {{Top.|X|J}} and {{Top.|Y|K}} if the topology isn't obvious) are ''homeomorphic''<ref name="ITTMJML"/> or ''topologically
  5 KB (731 words) - 22:58, 22 February 2017
• Index of notation
  ...ous map|continuous]] and have continuous derivatives up to (and including) order <math>k</math><br/> * Topology
  9 KB (1,490 words) - 06:13, 1 January 2017
• Passing to the quotient (function)
  ...rom [[passing to the quotient (topology)]] which is defined by Mond (2013, Topology) and Lee (Intro to Top manifolds), by further abstracting the claim</ref>: ...t is easy to see that we require {{M|1=[w(x)=w(y)]\implies[f(x)=f(y)]}} in order to proceed.
  8 KB (1,644 words) - 20:49, 11 October 2016
• Types of set algebras
  ...rently being refactored after being unchanged for more than 14 months. the order of elements and subheadings are likely to change and elements will be moved ...dition we can use the [[Borel sigma-algebra|Borel {{Sigma|algebra}}]] on a topology to get a {{Sigma|algebra}}, the below diagram is more complete, at the cost
  3 KB (449 words) - 20:06, 19 August 2016
• Classes of continuously differentiable functions
  # All [[Partial derivative|partial derivatives]] of {{M|f}} of order {{M|\le k}} exist and | Here we take it as the class of functions whose ''zeroth-order'' partial derivatives exist and are continuous<br/>This is simply the funct
  3 KB (632 words) - 20:32, 16 October 2015
• Site projects:Refactoring subjects
  ...to be consistently browsable. The user that spends a lot of time in the [[Topology (subject)]] part of the wiki must not get lost when heading into [[Abstract ...ect targeted at the uniformed reader. So introduce the topics in a logical order with things further into the page depending on those previously declared.
  3 KB (469 words) - 11:31, 19 February 2016
• Site:Mathematical subject index/Index
  ** [[Order Theory (subject)|Order Theory]] (see Order Theory for branches) * [[Algebraic Topology (subject)|Algebraic Topology]]
  2 KB (265 words) - 14:53, 26 February 2016
• Upper section
  * [[Lower section]] - the [[dual (order theory)|dual]] concept to this ** Closely related to the [[Alexandroff topology]]
  1 KB (171 words) - 16:35, 20 February 2016
• Index of operators on ordered sets
  The following are related more by topic than by any sort of other order (eg alphabetical) this is because the index ought to be quite short, and th ...roff topology]] of a [[preset]], where we get a [[topological space]]; the topology on {{M|\Uparrow\!\!A}} is {{M|\Upsilon\!A}} - the family of all [[upper sec
  2 KB (304 words) - 17:01, 20 February 2016
• Notes:Quotient topology plan
  ...' is the way forward, however the order in which to introduce the quotient topology, quotient space and quotient map can be varied. It's also not as if the con ==Quotient map results from applying the quotient topology==
  760 B (125 words) - 00:32, 22 April 2016
• Topological vector space
  ...rs]], so {{M|\mathbb{K}:\eq\mathbb{C} }} and let {{M|\mathcal{J} }} be a [[topology]] on {{M|X}} so that {{Top.|X|J}} is a [[topological space]]. We call the [ ...ne it this way for it topology first as in "topological vector space". The topology is "more implicit" when we speak of {{M|X}} than the field of a vector spac
  2 KB (383 words) - 14:03, 16 February 2017
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
  *** As {{M|\mathcal{J} }} is a [[topology]] we know it is closed under arbitrary [[union]] (the union of absolutely a ...on]] of any collection of open sets is an open set (by the definition of [[topology]]) we see:
  8 KB (1,529 words) - 00:27, 6 September 2016
• The canonical injections of the disjoint union topology are topological embeddings
  ...this construction we get some [[canonical injections of the disjoint union topology|canonical injections]]: ...{M|i_\beta}} - covered on the [[canonical injections of the disjoint union topology]] page so not shown on this pag
  3 KB (605 words) - 12:34, 26 September 2016
• Doctrine:Homotopy terminology
  ...entiable functions]], {{M|C^1}} means all continuous functions whose first-order [[partial derivatives]] are continuous, {{M|C^2}} means continuous with con {{Doctrine Of|Topology|Homotopy Theory}}
  3 KB (535 words) - 09:01, 31 October 2016
• Exercises:Mond - Topology - 1/Question 6
  {{float-right|{{Exercises:Mond - Topology - 1/Pictures/Q6P1 - 1}}}}We shall define {{M|f:[-1,1]\rightarrow\mathbb{S}^ ...-1\sim 1}}, use part 1 above and applying the ''[[passing to the quotient (topology)|topological version]]'' of [[passing to the quotient]] to find a ''[[conti
  7 KB (1,326 words) - 12:26, 12 October 2016
• Exercises:Mond - Topology - 1/Question 7
  {{float-right|{{Exercises:Mond - Topology - 1/Pictures/Q7 - 1}}}} ...ce]], with this definition we get a [[canonical projection of the quotient topology|canonical projection]], {{M|\pi:D^2\rightarrow D^2/\sim}} given by {{M|\pi:
  9 KB (1,732 words) - 23:26, 11 October 2016
• Proof that the fundamental group is actually a group/Outline
  ...end-point-preserving homotopy]] on {{C(I,X)}} - the set of all {{link|path|topology|s}} in {{M|X}} - but considered only on the subset of {{M|C([0,1],X)}}, {{M In order to factor {{M|(\pi\circ *)}} through {{M|(\pi,\pi)}} we require (as per the
  3 KB (454 words) - 18:31, 4 November 2016
• Exercises:Saul - Algebraic Topology - 1/Exercise 1.2
  ...^\circ}}, this is a homeomorphism. Some vertical stretching may also be in order, but that is also a homeomorphism. ====[[Exercises:Saul - Algebraic Topology - 1/Exercise 1.2/Lemmas|Lemmas]]====
  17 KB (3,132 words) - 12:03, 18 January 2017