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  • Injection
    An injective function is 1:1, but not nessasarally [[Surjection|onto]]. For a [[Function|function]] <math>f:X\rightarrow Y</math> every element of <math>X</math> is mapped t
    3 KB (463 words) - 21:50, 8 May 2018
  • Continuous map
    * The [[pre-image]] of every set open in {{M|Y}} under {{M|f}} is open in {{M|X}} ...,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a point {{M|x_0\in X}}, a [[function]] {{M|f:X\rightarrow Y}} is said to be ''continuous at {{M|x_0}}'' if<ref n
    6 KB (972 words) - 01:44, 14 October 2016
  • Basis for a topology
    * Need to add: [[A function is continuous if and only if the pre-image of every basis element is open]] - [[User:Alec|Alec]] ([[User talk:Alec|tal
    5 KB (802 words) - 18:35, 17 December 2016
  • Norm
    ...than this the references are valid, note that this is 'obvious' as if the image of {{M|\Vert\cdot\Vert}} could be in {{M|\mathbb{C} }} then the {{M|\Vert x Such a vector space equipped with such a function is called a [[Normed space|normed space]]<ref name="APIKM"/>
    6 KB (1,026 words) - 20:33, 9 April 2017
  • Basis and coordinates
    ![[Image:Coordinate example.JPG|500px]] Yes that function <math>L(x,y)\mapsto(2x,2y)'</math> is actually <math>[Id]_S^{S'}</math> thi
    9 KB (1,525 words) - 16:30, 23 August 2015
  • Measurable map
    '''Note: ''' Sometimes called a ''measurable [[Function|fuction]]''<ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref> ...ble if and only if for some generator of B (call it G) we have the inverse image of S is in A for every S in G|A map, {{M|f:(A,\mathcal{A})\rightarrow(F,\ma
    5 KB (792 words) - 02:31, 3 August 2015
  • Curve
    A parametrisation of a curve in {{M|\mathbb{R}^n}} is a function<ref> A parametrisation whos image is all (or a part of) a level curve is called a parameterisation (of part)
    1 KB (224 words) - 21:30, 28 March 2015
  • Compact-to-Hausdorff theorem
    Given a [[Continuous map|continuous]] and [[Bijection|bijective]] function between two [[Topological space|topological spaces]] {{M|f:X\rightarrow Y}} ::* (Using [[Image of a compact set is compact]])
    1 KB (219 words) - 12:36, 13 August 2015
  • Associative
    In fact given a function {{M|\times:S\times S\rightarrow S}} we even call the image of {{M|(x,y)}} under {{M|\times}} the ''product'' (or indeed the ''sum'' if
    455 B (77 words) - 07:44, 27 April 2015
  • Passing to the quotient (function)
    ...tyle=max-width:20em;}}Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}}<ref group="Note">I have chosen {{M|W}} to mean "wha ...P}(A)}} (a subset of {{M|A}}) we use {{M|f(D)}} to denote the {{link|image|function}} of {{M|D}} under {{M|f}}, namely: {{M|1=f(D):=\{f(d)\in B\ \vert\ d\in D\
    8 KB (1,644 words) - 20:49, 11 October 2016
  • Site projects:Patrolling measure theory/Task list
    * [[Additive function]] '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:50, 19 March 2 * [[Integral of a simple function (measure theory)]] '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]])
    5 KB (645 words) - 11:40, 21 August 2016
  • Topological property theorems
    * Image of a compact space is compact * A continuous and bijective function from a compact space to a Hausdorff space is a homeomorphism
    3 KB (616 words) - 08:37, 1 July 2015
  • Every map from a space with the discrete topology is continuous
    * Any [[Function|mapping]], {{M|1=f:X\rightarrow Y}}, is [[Continuous map|continuous]] ...thcal{K}[f^{-1}(A)\in\mathcal{J}]}} - for all open sets in {{M|Y}} the pre-image under {{M|f}} is open in {{M|X}}
    3 KB (534 words) - 13:07, 19 February 2016
  • Conditions for a map to be a measurable map
    This page is intended to be a list of different conditions for a [[Function|function]] to be a [[Measurable map|measurable map]] ...ble if and only if for some generator of B (call it G) we have the inverse image of S is in A for every S in G|A map, {{M|f:(A,\mathcal{A})\rightarrow(F,\ma
    678 B (120 words) - 23:54, 2 August 2015
  • Notes:Spherical coordinates
    * A chart is a function, {{M|\phi:U\rightarrow\mathbb{R}^n}} where {{M|U}} is some portion of our m * {{M|\text{Im}_\phi(U\cap V)}} is all the "image" of {{M|U\cap V}} under {{M|\phi}} and is the shaded bit on the bottom left
    10 KB (1,899 words) - 18:48, 23 September 2015
  • Motivation for smooth structures
    Suppose we have a [[topological manifold]] {{M|M}} and a function {{M|f:M\rightarrow\mathbb{R} }} which is continuous. Notice that the notion (In this article {{M|A_\varphi}} will denote the image of {{M|A}} under the chart {{M|\varphi}})
    2 KB (414 words) - 12:26, 12 November 2015
  • Exists functor
    ...on that takes each element of the powerset of the domain of {{M|f}} to the image of that element under {{M|f}} (which is obviously in the powerset of the co ...Describe this, something like {{M|b}} is in the image of {{M|X}} under the image of {{M|f}} in the functor ''if and only if'' there is an {{M|a\in X}} such
    2 KB (317 words) - 17:51, 13 March 2016
  • Notes:Continuous at a point
    ...nt EVERY open set containing the image of that point must have an open pre-image. :Indeed, the pre-image need not be open (otherwise it is about continuity everywhere, not just at
    1 KB (238 words) - 20:15, 23 March 2016
  • Site projects:Measure theory plan/Progress
    * [[Pre-image sigma-algebra]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) * [[Pre-image sigma-algebra/Definition]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:A
    2 KB (212 words) - 00:09, 2 April 2016
  • Notes:Basis for a topology/McCarty
    Suppose {{M|f:X\rightarrow Y}} is a [[function]] between two {{plural|topological space|s}}, {{Top.|X|J}} and {{Top.|Y|K}} {{Todo|1=I really need to create the pages that show the pre-image of functions preserves things like unions and intersections}}
    3 KB (467 words) - 16:58, 16 August 2016
  • Partial-function
    ...en a partial ordering is to a total ordering as a partial function is to a function}} Suppose {{M|f:A\rightarrow B}} is a ''partial [[function]]'', considering {{M|f}} as a [[relation]] this means that, for some {{M|a\
    2 KB (462 words) - 22:26, 23 August 2016
  • The canonical injections of the disjoint union topology are topological embeddings
    ...meomorphism]] between {{M|X_\beta}} and {{M|i_\beta(X_\beta)}} (its {{link|image|map}})) ...orphism]] onto (in the [[surjective]] sense of the word "onto") its {{link|image|map}}.
    3 KB (605 words) - 12:34, 26 September 2016
  • Every injection yields a bijection onto its image
    ...n by {{M|\overline{f}:x\mapsto f(x)}}- where {{M|f(X)}} denotes the {{link|image|map}} of {{M|X}} under {{M|f}}<ref name="imageDef" group="Note" /><!--body {{Function terminology navbox|plain}}
    4 KB (813 words) - 11:53, 26 September 2016
  • Exercises:Mond - Topology - 1/Question 6
    ***** By the nature of {{M|f}} being a [[function]] (only associating an element of the domain with one thing in the codomain #*# Note that (from {{link|passing to the quotient|function}}) that if {{M|f}} is surjective, then the resulting {{M|\fbar}} is surject
    7 KB (1,326 words) - 12:26, 12 October 2016
  • Exercises:Mond - Topology - 1/Question 7
    * Lastly, we define {{M|f:D^2\rightarrow\mathbb{S}^2}} to be the [[function composition|composition]] of {{M|E}} and {{M|f'}}, that is: {{M|1=f:=f'\cir ***** If {{M|1=x=y}} then by the nature of {{M|f}} being a [[function]] we must have {{M|1=f(x)=f(y)}}, we're done in this case
    9 KB (1,732 words) - 23:26, 11 October 2016
  • Exercises:Mond - Topology - 1/Question 8
    </noinclude>Suppose that {{M|f:X\rightarrow Y}} is a [[continuous function]] that is also [[surjective]], and let {{M|\sim}} denote the ''[[equivalenc ...{M|X}} is [[compact]], then {{M|\frac{X}{\sim} }} is compact also as [[the image of a compact set is compact]] and {{M|\pi:X\rightarrow\frac{X}{\sim} }} is
    3 KB (413 words) - 00:13, 12 October 2016
  • Properties of the pre-image of a map
    #REDIRECT [[Properties of the pre-image of a function]]
    103 B (15 words) - 00:28, 14 October 2016
  • A map is continuous if and only if the pre-image of every closed set is closed
    ===Continuity {{M|\implies}} the pre-image of every closed set is closed=== *** But by [[properties of the pre-image of a function]] we have {{M|1=f^{-1}(A-B)=f^{-1}(A)-f^{-1}(B)}} so:
    2 KB (378 words) - 01:39, 14 October 2016
  • Saturated set with respect to a function
    Let {{M|X}} and {{M|Y}} are [[sets]] and let {{M|f:X\rightarrow Y}} be any [[function]] between them. A [[subset of]] {{M|X}}, {{M|U\in\mathcal{P}(X)}}, is said ...a subset of {{M|Y}}, {{M|V}}, such that {{M|V}} is exactly the {{link|pre-image|map}} of {{M|U}} under {{M|f}}
    785 B (138 words) - 13:08, 16 October 2016
  • Invariant of an equivalence relation
    ...[[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from {{M|S}} to {{M|W}}. Then{{rALASR}}: ...'re saying the {{M|a\sim b}} property holds (doesn't vary) "under" (think "image of {{M|A}} under {{M|f}}"-like terminology) {{M|f}}, that {{M|1=f(a)=f(b)}}
    3 KB (478 words) - 18:58, 9 November 2016
  • Symmetric group
    * The group operation is ordinary function composition, for {{M|\sigma,\tau\in S_k}} we define: ...ain of {{M|\sigma}} considered as a [[function]] and thing below it is the image of that element under {{M|\sigma}}
    3 KB (425 words) - 12:21, 30 November 2016
  • A function is continuous if and only if the pre-image of every basis element is open
    * by [[properties of the pre-image of a function]], the pre-image of a union is the union of pre-images. *** But by [[properties of the pre-image of a function]] we see:
    4 KB (839 words) - 18:35, 17 December 2016
  • Exercises:Saul - Algebraic Topology - 3/Exercise 3.2
    Okay now {{M|\partial_1:\Delta_1(X)\rightarrow\Delta_0(X)}}, what is its image? This shows us that (sort of anyway) the image is spanned by various {{M|\text{Terminal}(\sigma_\alpha)-\text{Initial}(\si
    13 KB (2,312 words) - 06:33, 1 February 2017
  • Pre-image (map)
    #REDIRECT [[Pre-image (function)]]
    82 B (9 words) - 18:31, 31 January 2017
  • Pre-image
    * [[Pre-image (function)]] - {{M|f^{-1}(A)}} for a [[function]], {{M|f}} and a set {{M|A}} * [[Pre-image (relation)]] - of which the pre-image for a function is an instance
    228 B (35 words) - 22:28, 31 January 2017
  • Notes:Delta complex/Formal attempt
    ...athbb{N}_0}}) we don't need all of them, that statement 2 of the {{M|G_f}} function definition ensures the result is consistent. It's pretty obvious but I'm no ...ar map is injective if and only if its kernel is trivial]]''" and "''[[the image of a linear map is a vector subspace of the codomain]]''" wee see that:
    5 KB (966 words) - 14:36, 6 February 2017
  • Convex function
    ...:S\rightarrow\mathbb{R} }} be a [[function]]. We say {{M|f}} is a ''convex function'' if both of the following hold{{rAFCIRAPM}}: # The image of a point {{M|t}}-far along the line {{M|[x,y]}} is {{M|\le}} the point {{
    1 KB (224 words) - 10:54, 10 February 2017
  • Intermediate value theorem
    ...{{M|f:X\rightarrow}}[[the real line|{{M|\mathbb{R} }}]] is a [[continuous function]], then{{rITTMJML}}: ***#* By [[the image of a connected topological space is connected]] we see that, as {{M|X}} is
    5 KB (979 words) - 17:35, 17 February 2017
  • Locally Euclidean topological space
    *** As [[open balls]] are [[open sets]] and "''[[the pre-image of an open set under a homeomorphism is open]]''" we see: ...a(\varphi'(p);\mathbb{R}^n)\big)}} - by the definition of {{link|pre-image|function}}
    4 KB (667 words) - 14:32, 20 February 2017
  • Exercises:Saul - Algebraic Topology - 8/Exercise 8.5
    ...iven a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself]]'' ...iven a homeomorphism all subspaces of the domain are homeomorphic to their image under the homeomorphism itself/Statement}}
    7 KB (1,330 words) - 15:25, 7 March 2017
  • Local homeomorphism
    ...\text{Im}:A\rightarrow f(A)}} is the [[restriction onto its image]] of a [[function]]. ...{{M|f(U)}} is open in {{M|Y}} and {{M|f}} restricted to {{M|U}} (onto the image of {{M|U}}) is a [[homeomorphism]] (when {{M|U}} and {{M|f(U)}} are conside
    2 KB (271 words) - 21:45, 22 February 2017
  • Evenly covered by a continuous map
    ...I}V_\alpha}} - the [[union]] of this family is the entire {{link|pre-image|function}} of {{M|U}}<ref group="Note">Note that both sides of the {{M|\eq}} are ope ...text{ via }f\vert_{V_\alpha}^\text{Im}\big]}} - the [[restriction onto its image]] of {{M|f}} for each {{M|V_\alpha}} is a [[homeomorphism]] onto {{M|U}}
    2 KB (394 words) - 21:54, 24 February 2017
  • Axiom schema of replacement
    ...ter|FOL}} (or a [[parameter pack (FOL)|parameter pack]]) and describes a [[function between classes]]. ...[class function]] then for all sets {{M|X}}, {{M|F(X)}} - the {{link|image|function}} of {{M|X}} under {{M|F}} - denoted {{M|F(X)}} is also a set{{rSTTJ}}.
    2 KB (390 words) - 15:28, 5 April 2017
  • Poisson distribution
    |image=file.png |label2=[[Probability mass function|p.m.f]]
    8 KB (1,401 words) - 00:52, 20 July 2018
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