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  • Main Page
    ...t interface by far is the search box''' this project is intended to create a ''queriable resource'' for looking up things and learning from them (and th Some pages may be rather sparse, containing little more than a definition, but there is always some information even if it is minimal. Thi
    7 KB (999 words) - 16:51, 11 May 2020
  • Topology
    {{Stub page|grade=A*|msg=Should be easy to flesh out, find some more references and demote to g ...but rather a [[topological space]] which is a topology with its underlying set. See that page for more details}}
    3 KB (543 words) - 09:28, 30 December 2016
  • Power set
    ...as notation. This comes from the [[Cardinality|cardinality]] of the power set being <math>2^{|X|}</math>) ...haracteristic property of the power set is that <math>\forall U\subset X:U\in\mathcal{P}(X)</math>
    492 B (92 words) - 16:30, 23 August 2015
  • Injection
    {{Requires work|grade=A* |msg=This needs to be modified (in tandem with [[Surjection]]) to:
    3 KB (463 words) - 21:50, 8 May 2018
  • Topological space
    {{Stub page|grade=A|msg=Hasn't been updated since March 2015, in April 2016 it was updated to modern format and cleaned up}} ...hat {{M|\mathcal{J}_1\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_1[S\in\mathcal{J}_2]}}
    2 KB (268 words) - 13:37, 20 April 2016
  • Open set
    In a [[topological space]] {{M|(X,\mathcal{J})}} we have: ...J} }} that {{M|S}} is an open set. {{M|\mathcal{J} }} is by definition the set of open sets of {{M|X}}
    4 KB (677 words) - 02:26, 29 November 2015
  • Open ball
    Given a [[metric space]] {{M|(X,d)}} the ''open ball centred at {{M|x_0\in X}} of radius {{M|r>0}}'', denoted {{M|B_r(x_0)}} (however many notations a ...{x\in X\vert\ d(x,x_0)<r\} }} - that is all the points of {{M|X}} that are a distance (given by {{M|d}}) strictly less than {{M|r}} from {{M|x_0}}
    4 KB (842 words) - 02:00, 29 November 2015
  • Metric space
    A [[Normed space|normed space]] is a special case of a metric space, to see the relationships between metric spaces and others see ==Definition of a metric space==
    2 KB (336 words) - 06:07, 27 November 2015
  • Limit point
    ...of <math>x</math> has a non-empty [[Intersection|intersection]] with <math>A</math> that contains some point other than <math>x</math> itself. <math>x</math> is a limit point of <math>A</math> if <math>x\in\text{Closure}(A-\{x\})</math> (you can read about [[Closure, interior and boundary#Closure|
    877 B (133 words) - 14:09, 16 June 2015
  • Continuous map
    {{Refactor notice|grade=A}} ...are a few different conditions for continuity, there's also continuity at a point. This diagram is supposed to show how they relate to each other.
    6 KB (972 words) - 01:44, 14 October 2016
  • Surjection
    {{Requires work|grade=A*|msg=See [[Injection]]'s requires-work box [https://wiki.unifiedmathematics ...element of <math>B</math> is mapped onto from at least one thing in <math>A</math>
    2 KB (263 words) - 21:56, 8 May 2018
  • Dual vector space
    Given a [[Vector space|vector space]] {{M|(V,F)}} we define the '''dual''' or '''co * <math>V^*=\text{Hom}(V,F)</math> (recall this the set of all [[Homomorphism|homomorphisms]] (specifically [[Linear map|linear one
    3 KB (614 words) - 05:35, 8 December 2016
  • Connected (topology)
    {{Refactor notice|grade=A|msg=Ancient page, needs an update, linking to theorems, so forth}} Let {{Top.|X|J}} be a [[topological space]]. We say {{M|X}} is ''connected'' if{{rITTMJML}}:
    5 KB (866 words) - 01:52, 1 October 2016
  • Compactness
    ...I think there's a different definition for metric spaces, I have not seen a proof that the metric one {{M|\implies}} this one ...e may not speak of the compactness of subsets. Compactness is ''strictly'' a property of topological spaces.
    5 KB (828 words) - 15:59, 1 December 2015
  • Subspace topology
    {{Refactor notice|grade=A|msg=Needed urgently, ready to plough on with it now though!}} ...y empty, possibly equal to {{M|X}} itself</ref> be given. We can construct a new topological space, {{M|(S,\mathcal{J}_S)}} where the [[topology]] {{M|\
    6 KB (1,146 words) - 23:04, 25 September 2016
  • Quotient topology
    * Need to add [[Equivalent conditions to a map being a quotient map]] {{Refactor notice|grade=A|msg=This page is an embarrassment}}
    5 KB (795 words) - 13:34, 16 October 2016
  • Motivation for topology
    ...from this definition. The [[Open ball|open balls]] are open sets, any open set is the union of open balls. Can we go further? <math>\forall\text{open sets}\in Y,\ f^{-1}(\text{that open set})</math> is open. This looks very different from the definition.
    1 KB (243 words) - 15:39, 13 February 2015
  • Continuity definitions are equivalent
    ...th> being continuous (where the topologies are those [[Topology induced by a metric|induced by the metric]] are the same, that is ...all\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))</math>
    2 KB (476 words) - 07:20, 27 April 2015
  • The relationship between logical implication and the subset relation
    ...h> (and we say "A is a subset of B") if and only if every element of <math>A</math> also belongs to <math>B</math> ...[x\in A\implies x\in B]</math><ref>Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jec
    776 B (136 words) - 17:36, 16 January 2017
  • Basis for a topology
    {{Refactor notice|grade=A|msg=Updating with findings. * Need to add: [[A function is continuous if and only if the pre-image of every basis element
    5 KB (802 words) - 18:35, 17 December 2016
  • The intersection of sets is a subset of each set
    That <math>A\cap B\subset A</math> ...h>A\cap B\subset B</math> (as <math>A\cap B=B\cap A</math> and <math>B\cap A\subset B</math> by the statement above)<ref>Alec's (my) own work</ref>
    820 B (159 words) - 21:35, 15 August 2015
  • Commutativity of intersection
    <math>A\cap B=B\cap A</math> This is somewhere between a theorem and a definition because at some point you have to accept "and" is commutative or
    610 B (122 words) - 19:32, 28 October 2016
  • Closed set
    ...space]] <math>(X,\mathcal{J})</math> is a set <math>A</math> where <math>X-A</math> is open<ref>Introduction to topology - Third Edition - Mendelson</re ...y [[metric space]] is also a [[topological space]] it is still true that a set is closed if its complement is open.
    1 KB (238 words) - 15:36, 24 November 2015
  • Vector space
    A vector space {{M|V}} over a [[Field|field]] {{M|F}} is a non empty set {{M|V}} and the binary operations: Such that the following 8 "axioms of a vector space" hold
    2 KB (421 words) - 16:30, 23 August 2015
  • The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
    ...n Apr 2015 when [[Template:Theorem]] was moved to [[Template:Theorem Of]]. A very old page indeed!}} ...et {{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }} be a [[metric]] on that set and let {{M|(X,d)}} be the resulting [[metric space]]. Then we claim:
    4 KB (814 words) - 22:16, 16 January 2017
  • Closure, interior and boundary
    ...} }} ("bar" in LaTeX) or {{M|\overline{A} }} ("overline" in LaTeX) is the set:<ref>Introduction to Topological Manifolds - John Lee</ref> ...h>\overline{A}=\bigcap\{B\subset X|A\subset B\text{ and }B\text{ is closed in }X\}</math>
    1 KB (210 words) - 00:20, 9 March 2015
  • Measure space
    ...ame="MIM">Measures, Integrals and Martingales - Rene L. Schilling</ref> is a [[Tuple|tuple]]: * {{M|(X,\mathcal{A},\mu:\mathcal{A}\rightarrow[0,+\infty])}} - but because [[Mathematicians are lazy]] we simp
    1 KB (188 words) - 15:24, 21 July 2015
  • Triangle inequality
    ...[Metric space|metric space]]) of which <math>|x-z|\le|x-y|+|y-z|</math> is a special case. ...way of writing it is <math>|a+b|\le |a|+|b|</math>, notice if we set {{M|1=a=x-y}} and {{M|1=b=y-z}} then we get <math>|x-y+y-z|\le|x-y|+|y-z|</math> wh
    3 KB (546 words) - 13:05, 19 February 2016
  • Product topology
    {{Refactor notice|grade=A|As a part of the topology patrol}} {{Requires references|grade=A|msg=Check Munkres and Topological Manifolds}}
    5 KB (871 words) - 20:32, 23 September 2016
  • Tuple
    ...ever as it may not be in a [[Vector space|vector space]] we do not call it a vector. ...3,\{a\})</math> is the tuple of the numbers 1, 2 and 3 and the set <math>\{a\}</math>
    610 B (97 words) - 16:30, 23 August 2015
  • Function
    A function {{M|f}} is a special kind of [[Relation|relation]] ...]<ref name="API">Analysis - Part 1: Elements - Krzysztof Maurin</ref>, for a relation:
    4 KB (659 words) - 13:01, 19 February 2016
  • Set theory axioms
    ...sed to provide some discussion for the axioms (for example "there exists a set with no elements" doesn't really deserve its own page) ...lt. Only "major" results are shown, they are covered in the motivation for set theory page, and "D" denotes "definition" - which is something that is natu
    3 KB (619 words) - 10:25, 11 March 2015
  • Motivation for set theory axioms
    ...s. For example you are happy that {{M|\{1,2,3\}=\{2,1,3\} }} and that this set (I use singular because they are identical) contains the elements {{M|1}}, ...- for example we use a letter (<math>\emptyset</math>) to denote the empty set, what if there are 2 empty sets? We would like to justify this.
    3 KB (584 words) - 23:03, 28 February 2015
  • Relation
    ...should be assumed if just relation is specified</ref>) between two sets is a subset of the [[Cartesian product]] of two sets{{rAPIKM}}<ref name="TAPL">T We say that {{M|\mathcal{R} }} is a ''relation in {{M|X}}''<ref name="APIKM"/> if:
    4 KB (762 words) - 20:07, 20 April 2016
  • Ordered pair
    ...dered pair <math>(a,b)=\{\{a\},\{a,b\}\}</math>, this way <math>(a,b)\ne(b,a)</math>. Ordered pairs are vital in the study of [[Relation|relations]] which leads to [[Function|functions]]
    2 KB (327 words) - 07:22, 27 April 2015
  • The domain, range and field of a relation exist
    It is important to know that the domain and range of a [[Relation|relation]] exist. ...math>\{x|\exists y:(x,y)\in R\}</math> exists, to do this we require the [[Set theory axioms|axioms]] of schema and union.
    339 B (63 words) - 07:22, 27 April 2015
  • Ordering
    ...g|Proof that <math>\le</math> is a partial ordering <math>\iff <</math> is a strict ordering]] ...special kind of [[Relation|relation]], we can define an order uniquely as a partial or strict ordering. That is the two are equivalent.
    5 KB (1,006 words) - 13:21, 1 January 2016
  • Equivalence relation
    A [[relation]], {{M|\sim}}, in {{M|X}}<ref group="Note">This terminology means {{M|\sim \subseteq X\times | {{M|\forall x\in X[(x,x) \in \sim]}}. Which we write {{M|\forall x\in X[x\sim x]}}.
    3 KB (522 words) - 15:18, 12 February 2019
  • Linear map
    ...a u+\beta v)=\alpha T(u)+\beta T(v)\ \forall u,v\in V\ \forall\alpha,\beta\in F}} | A linear transform into the same space as the domain, that is {{M|T:(V,F)\rig
    3 KB (512 words) - 16:30, 23 August 2015
  • Index of notation
    ...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 any other ! [[Index of set-like notations]]
    9 KB (1,490 words) - 06:13, 1 January 2017
  • Span, linear independence, linear dependence, basis and dimension
    Given a set of vectors {{M|S}} in a vector space {{M|(V,F)}} ...n}(S)=\left\{\sum^n_{i=1}\lambda v_i|n\in\mathbb{N},\ v_i\in S,\ \lambda_i\in F\right\}</math>
    2 KB (330 words) - 18:07, 25 April 2015
  • Sequence
    A sequence is one of the earliest and easiest definitions encountered, but I ...prefer it. This notation is inline with that of a [[Tuple|tuple]] which is a generalisation of [[Ordered pair|an ordered pair]].
    2 KB (419 words) - 18:12, 13 March 2016
  • Basis and coordinates
    This is very much a "motivation" page and a discussion of the topic. ==What is a coordinate==
    9 KB (1,525 words) - 16:30, 23 August 2015
  • Formal logic language
    ...y so in set theory, but overall an important role. It is important to have a concrete understanding of this. *<math>\text{Plus}(\text{Natural }a,\text{Natural }b)\rightarrow\text{Natural}</math>
    2 KB (410 words) - 16:35, 9 March 2015
  • Group
    {{Requires references|The bulk of this page was written when this was a 'note project' and was taken from books (even though I was already really f ...notes the group's operation applied to the elements {{M|a\in G}} and {{M|b\in G}}.
    7 KB (1,332 words) - 07:17, 16 October 2016
  • Cardinality
    Informally the cardinality of a set is the number of things in it. The cardinality of a set {{M|A}} is denoted <math>|A|</math>
    2 KB (327 words) - 10:25, 12 March 2015
  • Ring of sets
    A Ring of sets is also known as a '''Boolean ring''' ...sets]] is also a ring, and that an [[Algebra of sets]] is sometimes called a '''Boolean algebra'''
    2 KB (336 words) - 17:21, 18 August 2016
  • Algebra of sets
    : '''Note: ''' Every ''algebra of sets'' is a ''[[ring of sets]]'' (see below) An ''algebra of sets'' is a collection of sets, {{M|\mathcal{A} }} such that{{rMTH}}:
    3 KB (507 words) - 18:43, 1 April 2016
  • Sigma-ring
    A '''Sigma-ring''' or <math>\sigma</math>-ring, is closely related to [[Ring ...} is a {{sigma|ring}} if<ref>Measure Theory, p24 - Halmos - Graduate Texts in Mathematics (18) - Springer</ref>:
    728 B (125 words) - 15:34, 13 March 2015
  • Sigma-algebra
    ...: ''' A ''Sigma-algebra'' of sets, or {{sigma|algebra}} is very similar to a [[Sigma-ring|{{sigma|ring}}]] of sets. ...ts]] is to an [[algebra of sets]] as a [[sigma-ring|{{sigma|ring}}]] is to a ''{{sigma|algebra}}''
    8 KB (1,306 words) - 01:49, 19 March 2016

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