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• Factoring a function through the projection of an equivalence relation induced by that function yields an injection/Diagram
  261 B (38 words) - 22:54, 8 October 2016
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
  ...e relation]]'' [[equivalence relation induced by a function|induced by the function {{M|f}}]], recall that means: ...gh {{M|\pi:X\rightarrow \frac{X}{\sim} }}<ref group="Note">the [[canonical projection of the equivalence relation]], given by {{M|\pi:x\mapsto [x]}} where {{M|[x
  6 KB (1,097 words) - 20:24, 9 October 2016
• If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection
  ...function through the projection of an equivalence relation induced by that function yields an injection]]}} ...function through the projection of an equivalence relation induced by that function yields an injection]]
  2 KB (339 words) - 12:57, 9 October 2016

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• Equivalence relation
  * [[Canonical projection of an equivalence relation]] ...he quotient|function}} - things are often factored through the [[canonical projection of an equivalence relation]]
  3 KB (522 words) - 15:18, 12 February 2019
• 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
• Notes:Pick function
  Let {{M|P}} denote the {{M|4\times 4}} (projection AND view operator) you are using (that is for a point {{M|x}}, {{M|Px}} is ...x}} this matrix applies the camera position/angle transformation, then the projection operation all at once. Thus {{M|P}} represents the camera.
  4 KB (686 words) - 01:43, 15 September 2015
• Index of notation/P
  | [[Prime number|Prime numbers]], [[Projection (function)|projective functions]] (along with {{M|\pi}}), vector points (typically {{ * {{M|\mathbb{P} }} denotes a [[probability function]] almost always. However I have seen:
  1 KB (184 words) - 23:58, 11 January 2016
• Site projects:Patrolling topology/Task list
  * [[Continuous function]] * [[Natural projection]]
  4 KB (404 words) - 21:36, 30 September 2016
• Notes:Quotient topology
  </div>This is very similar to [[quotient (function)|the quotient of a function]].<br/> ...us {{M|\iff}} {{M|f\circ\pi}} is continuous ({{M|\pi}} being the canonical projection)
  6 KB (1,087 words) - 19:45, 26 April 2016
• Characteristic property of the product topology/Statement
  ...}} (where {{M|\pi_\alpha}} denotes the {{M|\alpha^\text{th} }} [[canonical projection of the product topology]]) then: ...ta:Y\rightarrow X_\beta\text{ is continuous}]}} - in words, each component function is continuous
  2 KB (340 words) - 20:55, 23 September 2016
• Disjoint union topology
  ...nical projection" FROM the product to the spaces, as this would not be a [[function]]!</ref> - be sure to notice the abuse of notation going on here.
  2 KB (316 words) - 23:47, 25 September 2016
• Notes:Group isomorphism theorems
  ...nel (group theory)|kernel]] contains {{M|N}}, then {{M|\varphi}} [[factor (function)|factors]] uniquely through {{M|\pi:G\rightarrow G/N}}. In line with the [[factor (function)]] page, note that we require {{M|1=\forall x,y\in G[(\pi(x)=\pi(y))\implie
  4 KB (654 words) - 23:06, 10 July 2016
• Group factorisation theorem
  ...he [[group]] {{M|B}}.</ref> and {{M|\pi:G\rightarrow G/N}} the [[canonical projection of the quotient group]], let {{M|\varphi:G\rightarrow H}} be any [[group ho ...instead of {{M|\varphi(g)}}!", these are actually the same, see [[Factor (function)]] for more details, I shall explain this here.
  7 KB (1,195 words) - 22:55, 3 December 2016
• An injective group homomorphism means the group is isomorphic to its image
  ...lready, {{M|\pi:A\rightarrow A/\text{Ker}(\varphi)}} (see: the [[canonical projection of the quotient group]] for details about {{M|\pi}), we only need to show i ...(a)=\pi(b)\implies\varphi(a)=\varphi(b)}} in order to be able to [[factor (function)|factor]]. We can use that again here.
  4 KB (727 words) - 04:53, 20 July 2016
• Notes:Quotient topology/Table
  ...e relation]], {{M|\sim}} is involved then the "natural map" (''[[canonical projection of an equivalence relation]]'') is an '''identification''' ...let {{M|Y}} be a [[set]] and {{M|f:X\rightarrow Y}} a ''[[surjective]]'' [[function]]. The '''quotient [[topology]]''' on {{M|Y}} ({{AKA}}: ''topology induced
  2 KB (327 words) - 16:09, 13 September 2016
• Exercises:Mond - Topology - 1/Question 7
  ...inition we get a [[canonical projection of the quotient topology|canonical projection]], {{M|\pi:D^2\rightarrow D^2/\sim}} given by {{M|\pi:x\mapsto [x]}} where * 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
  9 KB (1,732 words) - 23:26, 11 October 2016
• Equivalence relation induced by a function
  ...that {{M|f}} may be [[factor (function)|factored]] through the [[canonical projection of an equivalence relation]] to yield an injection. Furthermore if {{M|f}} ...tion of the equivalence relation induced by that function then the yielded function is a bijection]]
  2 KB (276 words) - 22:40, 8 October 2016
• Canonical projection of an equivalence relation
  ...x]}} (where {{M|[x]}} denotes the [[equivalence class]] of {{M|x}}) is a [[function]]. We call this map the "''canonical projection of the equivalence relation''", sometimes just the "''canonical map of the
  697 B (107 words) - 22:33, 8 October 2016
• Factoring a function through the projection of an equivalence relation induced by that function yields an injection
  ...e relation]]'' [[equivalence relation induced by a function|induced by the function {{M|f}}]], recall that means: ...gh {{M|\pi:X\rightarrow \frac{X}{\sim} }}<ref group="Note">the [[canonical projection of the equivalence relation]], given by {{M|\pi:x\mapsto [x]}} where {{M|[x
  6 KB (1,097 words) - 20:24, 9 October 2016
• If a surjective function is factored through the canonical projection of the equivalence relation induced by that function then the yielded function is a bijection
  ...function through the projection of an equivalence relation induced by that function yields an injection]]}} ...function through the projection of an equivalence relation induced by that function yields an injection]]
  2 KB (339 words) - 12:57, 9 October 2016
• Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuous map
  ...function through the projection of an equivalence relation induced by that function yields an injection]]''" by simply considering [[continuity]] in addition. ...through the ''[[canonical projection of the equivalence relation|canonical projection]]'' of the ''[[equivalence relation]]'' [[equivalence relation induced by a
  3 KB (430 words) - 22:23, 9 October 2016
• If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a continuous bijection
  : '''Note: ''' "''[[Factoring a continuous map through the projection of an equivalence relation induced by that map yields an injective continuo ...]] through the [[canonical projection of an equivalence relation|canonical projection]] of the ''[[equivalence relation]]'' [[equivalence relation induced by a m
  2 KB (264 words) - 22:32, 9 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 * "''[[If a surjective continuous map is factored through the canonical projection of the equivalence relation induced by that map then the yielded map is a c
  3 KB (413 words) - 00:13, 12 October 2016