Search results

From Maths
Jump to: navigation, search

Create the page "Non-empty in a set" on this wiki! See also the search results found.

Page title matches

Page text matches

  • 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
  • Measure Theory
    ...}^n</math> means <math>[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\in\mathcal{J}^n</math> We can clearly get a ring from this, but not a [[Sigma-ring|{{Sigma|ring}}]] as for example:
    4 KB (733 words) - 01:41, 28 March 2015
  • Extended real value
    If a [[Function|function]] is described as "extended real valued" it means: <mat The set <math>\mathbb{R}\cup\{-\infty,+\infty\}</math> refers to "extended real val
    2 KB (396 words) - 16:07, 13 March 2015
  • Set function
    ...lass of sets<ref>Halmos - Measure Theory - p30 - Springer - Graduate Texts in Mathematics (18)</ref> * a [[Measure|measure]] is a good example (and indeed motivates the definition)
    355 B (54 words) - 16:10, 13 March 2015
  • Additive function
    ...ve function (which way have meaning in say algebra), be sure to update the SET FUNCTION redirects that point into this page ...function]] on a class of [[set|sets]], {{M|\mathcal{A} }}, {{M|f:\mathcal{A}\rightarrow\mathbb{R} }} is called ''additive'' or ''finitely additive'' if
    6 KB (971 words) - 18:16, 20 March 2016
  • Pre-measure
    ...ef> is a [[Measure|measure]] on an [[Algebra of sets|algebra]] rather than a [[Sigma-algebra|{{Sigma|algebra}}]], the properties are as follows: ...an algebra of sets (a system of subsets of {{M|X}}) and {{M|\mu_0:\mathcal{A}\rightarrow[0,+\infty]}} such that:
    5 KB (782 words) - 01:49, 26 July 2015
  • Measure
    {{Stub page|Requires further expansion|grade=A}}{{Extra Maths}}{{:Measure/Infobox}} A (positive) ''measure'', {{M|\mu}} is a [[set function]] from a [[sigma-ring|{{sigma|ring}}]], {{M|\mathcal{R} }}, to the positive [[extend
    6 KB (941 words) - 14:39, 16 August 2016
  • Measurable space
    {{Refactor notice|grade=A*|msg=Lets get this measure theory stuff sorted. At least the skeleton ...set]], {{M|X}}, and a [[sigma-algebra|{{sigma|algebra}}]], {{M|\mathcal{A}\in\mathcal{P}(\mathcal{P}(X))}}<ref group="Note">More neatly written perhaps:
    2 KB (248 words) - 13:05, 2 February 2017
  • Lebesgue measure
    ...half-open rectangle <math>[\![a,b)\!)=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}</math> as follows: <math>\lambda^n\big([\![a,b)\!)\big)=\prod^n_{i=1}(b_i-a_i)</math>
    804 B (129 words) - 00:28, 20 December 2016
  • Subgroup
    ...(G,\times_G:G\times G\rightarrow G)}} is a set {{M|H\subseteq G}} which is a [[Group|group]] under the operation {{M|\times_G}} restricted to {{M|H\time ...imes G\rightarrow G)}} we say {{M|(H,\times_H:H\times H\rightarrow H)}} is a subgroup of {{M|(G,\times_G)}} if:
    2 KB (364 words) - 17:35, 15 March 2015
  • Coset
    Let {{M|(G,\times)}} be a [[Group|group]] and {{M|(H,\times)}} a [[Subgroup|subgroup]] we denote cosets as follows: Given any <math>g\in G</math> the
    3 KB (616 words) - 18:08, 15 March 2015
  • Semi-ring of sets/Definition
    ...inished loop and the foot at the right. "Semi '''R'''ing".</ref> is called a ''semi-ring of sets'' if{{rMIAMRLS}}: # {{M|1=\emptyset\in\mathcal{F} }}
    2 KB (337 words) - 17:25, 18 August 2016
  • Ring generated by
    Here ring refers to [[Ring of sets|a ring of sets]] Given any class of sets {{M|A}}, there exists a unique ring {{M|R_0}} such that {{M|E\subseteq R_0}} and such that if {{M|R
    2 KB (307 words) - 07:24, 27 April 2015
  • Complement
    ...|X}} the complement of {{M|A}} (often denoted {{M|A^c}}, {{M|A'}} or {{M|C(A)}}) is given by: <math>A^c=\{x\in X|x\notin A\}=X-A</math>
    726 B (145 words) - 13:28, 18 March 2015
  • Ring
    {{Refactor notice|grade=A|msg=Page was ancient, mostly written in May 2015}} Not to be confused with [[Ring of sets|rings of sets]] which are a topic of [[Algebra of sets|algebras of sets]] and thus [[Sigma-algebra|{{Si
    7 KB (1,248 words) - 05:02, 16 October 2016
  • Monotonic set function
    Sometimes called '''Monotone set function'''. A set function {{M|f:E\rightarrow [0,\infty]\subset\mathbb{R} }} is '''monotonic'
    548 B (87 words) - 14:18, 18 March 2015
  • Subtractive set function
    ...sure Theory - Graduate Texts in Mathematics - 18</ref> if whenever {{M|A,B\in E}} we have <math>A\subseteq B\wedge B-A\in E\wedge |f(B)|<\infty\implies f(B-A)=f(B)-f(A)</math>
    409 B (71 words) - 14:24, 18 March 2015
  • Measures are monotonic and subtractive
    ...function|subtractive]]<ref>p37 - Halmos - Measure Theory - Graduate Texts in Mathematics - book 18</ref>
    318 B (41 words) - 07:26, 27 April 2015
  • Probability space
    ...at <math>\mu(X)=1</math> (and we will write {{M|\mu}} as {{M|\mathbb{P} }} in line with notation). Now {{M|(X,\mathcal{A},\mathbb{P})}} is a ''Probability space''
    2 KB (338 words) - 22:55, 2 May 2015
  • Tangent space
    | The set of tangents to a point in {{M|\mathbb{R}^n}}<br/> <math>G_p(\mathbb{R}^n)=\{(p,v)|v\in\mathbb{R}^n\}</math> - the set of all arrows at {{M|p}}
    6 KB (1,190 words) - 19:27, 14 April 2015
  • Manifolds
    ...ove around the surface as if it were a plane. Of course it is not actually a plane. ...[Chart|chart]] (sometimes a "coordinate chart"), it is a map, from an open set of the manifold {{M|U}} to some open subset of {{M|\mathbb{R}^n}}, for exam
    2 KB (276 words) - 05:59, 7 April 2015
  • Set of all derivations at a point
    '''NOTE:''' NOT to be confused with [[Set of all derivations of a germ]] Some authors use <math>T_p(\mathbb{R}^n)</math> to denote this set (the set of derivations of the form <math>\omega:C^\infty\rightarrow\mathbb{R}</math
    2 KB (291 words) - 21:51, 13 April 2015
  • Germ
    A germ is an [[Equivalence class|equivalence class]] of an [[Equivalence rela ...ine an equivalence relation on the <math>C^\infty</math> functions defined in some neighbourhood of {{M|p}} as:
    2 KB (285 words) - 01:36, 5 April 2015
  • The set of all germs of smooth functions at a point
    ...e assumed here (so <math>C^k_p(\mathbb{R}^n)</math> is a notation for this set, but there is no general one for other kinds of functions - use words to de * Assume "on <math>\mathbb{R}^n</math>" in the absence of a space. That is, assume {{M|C^\infty_p}} denotes {{M|C^\infty_p(\mathbb{R}^n
    794 B (140 words) - 01:50, 5 April 2015
  • Equivalence class
    Given an [[Equivalence relation]] {{M|\sim}} the equivalence class of {{M|a}} is denoted as follows: <math>[a]=\{b|a\sim b\}</math>
    1 KB (160 words) - 20:00, 14 November 2015
  • Derivation
    ...C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} }} is a '''''derivation at {{M|a}}''''' if it is [[Linear map|{{M|\mathbb{R} }}-linear and satisfies the fol * Given {{M|f,g\in C^\infty(\mathbb{R}^n)}} we have:
    2 KB (325 words) - 18:08, 14 October 2015
  • Chart
    '''Note:''' Sometimes called a coordinate chart ...on a [[Topological manifold|topological manifold]] of dimension {{M|n}} is a pair {{M|(U,\varphi)}}<ref>John M Lee - Introduction to smooth manifolds -
    2 KB (322 words) - 06:32, 7 April 2015
  • Open and closed maps
    A <math>f:(X,\mathcal{J})\rightarrow (Y,\mathcal{K})</math> (which need not b ...e image of an open set is open (that is <math>\forall U\in\mathcal{J}[f(U)\in\mathcal{K}]</math>)
    4 KB (692 words) - 08:00, 8 April 2015
  • Smooth function
    ...on a [[Smooth manifold|smooth {{n|manifold}}]], {{M|(M,\mathcal{A})}}, is a function<ref>Introduction to smooth manifolds - John M Lee - Second Edition ...ge f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty]}}
    3 KB (560 words) - 16:16, 14 April 2015
  • The fundamental group
    {{Refactor notice|grade=A|msg=I cannot believe it's been 15 months and this still isn't complete! Let {{Top.|X|J}} be a [[topological space]] {{M|\text{Loop}(X,b)\subseteq C(I,X)}} and consider t
    3 KB (393 words) - 16:10, 4 November 2016
  • Circle
    A circle is usually defined by {{M|1=\mathcal{S}^1=\Big\{(x,y)\in\mathbb{R}^2{{!}}d\Big((0,0),(x,y)\Big)=1 \Big\} }} ...{M|f:t\mapsto e^{2\pi jt} }} is significant. As it makes {{M|\mathbb{R} }} a [[Covering space|covering space]] of {{M|\mathbb{S}^1}}
    3 KB (592 words) - 16:57, 11 May 2015
  • Compact-to-Hausdorff theorem
    Given a [[Continuous map|continuous]] and [[Bijection|bijective]] function between * '''Then {{M|f}} is a [[Homeomorphism|homeomorphism]]'''<ref>Introduction to Topology - Nov 2013
    1 KB (219 words) - 12:36, 13 August 2015
  • Relatively open
    ...ely open'''<ref>Introduction to topology - Third Edition - Mendelson</ref> in {{M|X}} Alternatively we may say given a {{M|A\subseteq X}} the family of sets:
    592 B (97 words) - 18:42, 19 April 2015
  • Relatively closed
    ...ological space]] {{M|(X,\mathcal{J})}} if {{M|A}} is [[Closed set|closed]] in {{M|Y}} * [[Closed set]]
    406 B (60 words) - 18:40, 19 April 2015
  • Hausdorff space
    ...ce|review=true|msg=Page was 1 year and 1 day since modification, basically a stub, seriously needs an update. ...finite complement topology is not Hausdorff]] as an example of a familiar set with an unfamiliar topology
    4 KB (679 words) - 22:52, 22 February 2017
  • Addition of vector spaces
    ...ef>Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics</ref> <math>V=\mathop{\boxplus}^n_{i=1}V_i=\left\{(v_1,\cdots,v_n)|v_i\in V_i,\ i=1,2,\cdots,n\right\}</math><br/>
    4 KB (804 words) - 18:02, 18 March 2016
  • Monoid
    ...d Third Edition - Graduate Texts In Mathematics</ref> is a set {{M|S}} and a function {{M|\times_S:S\times S\rightarrow S}} (called the operation) such * [[Associative]] - that is <math>\forall x,y,z\in S[(xy)z=x(yz)]</math>
    735 B (131 words) - 07:48, 27 April 2015
  • Almost always
    ...'''Almost everywhere''' or '''Almost all'' are phrases that mean ''all but a finite number''<ref>Algebra - Serge Lang - Revised Third Edition - GTM</ref {{Todo|But in measure theory and probability it means all but a set of measure zero}}
    694 B (115 words) - 21:44, 19 March 2016
  • Surjection/Definition
    Given a [[Function|function]] {{M|f:X\rightarrow Y}}, we say {{M|f}} is ''surjectiv * <math>\forall y\in Y\exists x\in X[f(x)=y]</math>
    273 B (54 words) - 17:42, 10 May 2015
  • Kernel
    ...identity element), the kernel of {{M|f:X\rightarrow Y}} (where {{M|f}} is a [[Function|function]]) is defined as: * <math>\text{Ker}(f)=\{x\in X|f(x)=e\}</math> where <math>e</math> denotes the identity of {{M|Y}}
    2 KB (376 words) - 19:53, 10 May 2015
  • Commutator subgroup
    ...s)}}. Then {{M|C}} is a [[Subgroup|sugroup]] of {{M|G}}, furthermore it is a [[Normal subgroup|normal subgroup]]. That is to say: * <math>C=\langle\{[g,h]\in G\ |\ g,h\in G\}\rangle</math>
    394 B (65 words) - 11:21, 12 May 2015
  • Generated subgroup
    A [[Cyclic subgroup|cyclic subgroup]] is a group generated by a single element. Let {{M|(G,\times)}} be a group, and {{M|\{g_1,\cdots,g_n\}\subset G}} be a set of elements of {{M|G}}, then the ''subgroup generated by {{M|1=\{g_i\}_{i=1
    2 KB (404 words) - 12:39, 12 May 2015
  • Cyclic subgroup
    ...s a [[Generated subgroup|generated subgroup]], where the generating set is a single element of the [[Group|group]] {{M|(G,\times)}}, that is: * For any {{M|g\in G}} the ''cyclic subgroup generated by {{M|g}}'' is {{M|\langle g\rangle}}
    580 B (94 words) - 14:12, 12 May 2015
  • Greatest common divisor
    '''Note:''' requires knowledge of what it means for a number to be a [[Divisor|divisor]] of another. ...the greatest positive integer, {{M|d}}, that [[Divisor|divides]] both {{M|a}} and {{M|b}}. We write:
    1 KB (252 words) - 08:33, 21 May 2015
  • Span
    Given a set of vectors {{M|S}} in a [[Vector space|vector space]] {{M|(V,F)}} the '''span'''<ref>Advanced Linea ...\left\{\sum^n_{i=1}\lambda v_i\Big|\ n\in\mathbb{N},\ v_i\in S,\ \lambda_i\in F\right\}</math>
    1,013 B (173 words) - 17:09, 28 May 2015
  • Passing to the quotient (function)
    : See [[Passing to the quotient]] for a disambiguation of this term. {{float-right|{{/Diagram}}|style=max-width:20em;}}Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W
    8 KB (1,644 words) - 20:49, 11 October 2016
  • Indicator function
    ...ubseteq X}} and typically {{M|1=\alpha=0\in\mathbb{R} }} and {{M|1=\beta=1\in\mathbb{R} }} where<ref>Probability and Measure - Anniversary Edition - Patr 1 & x\in A\\
    600 B (91 words) - 02:26, 7 June 2015
  • Notes:ToMond
    I have phrased this a bit weirdly but that's to make it parallel to tensors, I also use {{M|W}} t ...ear. Here {{M|W}} is "whatever" and is a vector space. {{M|\tilde{f} }} is a linear map
    5 KB (921 words) - 05:43, 7 June 2015
  • Direct sum (ring)
    ...mes_R)}} and {{M|(S,+_S,\times_S)}} their ''direct sum'' is defined on the set {{M|R\times S}} (where {{M|\times}} is the [[Cartesian product]]), that is: * {{M|1=R\times S=\{(x,y)\vert\ x\in R\wedge y\in S\} }}
    3 KB (549 words) - 14:32, 8 June 2015
  • Ordered integral domain
    ...bstract Algebra - An Expanded Version - Neal H. McCoy</ref> if it contains a subset, which we'll denote {{M|D^+}} with the following properties: # {{M|a,b\in D^+\implies a+b\in D^+}} (closed under addition)
    1,016 B (168 words) - 16:10, 23 August 2015
  • Continuity and non-surjective functions
    ...gical space|path]] in a [[topological space]], {{M|(X,\mathcal{J})}}, then a path is simply: ...ot\vert)]}} where {{M|\mathcal{O} }} denotes the [[open set|open sets]] of a space
    3 KB (556 words) - 17:42, 6 September 2015

View (previous 100 | next 100) (20 | 50 | 100 | 250 | 500)

Retrieved from "https://wiki.unifiedmathematics.com/index.php?title=Special:Search"