Difference between revisions of "Notes:Quotient topology"
From Maths
m |
m (→According to John M. Lee) |
||
| (7 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | '''Note to readers: ''' the page [[quotient topology]] as it stands right now ( | + | : '''Note to readers: ''' the page [[quotient topology]] as it stands right now ([[User:Alec|Alec]] ([[User talk:Alec|talk]]) 17:07, 21 April 2016 (UTC)) is an embarrassment to me. However before I can clean it up I must unify it. I've been using it for almost 2 years now though I promise! Gosh this is embarrassing. |
| + | |||
| + | |||
| + | See [[Notes:Quotient topology plan]] for an outline of the page. | ||
| + | __TOC__ | ||
==According to John M. Lee== | ==According to John M. Lee== | ||
| + | : '''Addendum: ''' for some reason I lie here, the author considers a map {{M|f:X\rightarrow A}} where {{M|A}} is a [[set]] and later applies the equivalence relation version to it. | ||
Let {{M|\sim}} denote an [[equivalence relation]], let {{M|(X,\mathcal{J})}} be a [[topological space]]. We get a map, {{M|\pi:X\rightarrow\frac{X}{\sim} }} that takes {{M|\pi:x\rightarrow[x]}} | Let {{M|\sim}} denote an [[equivalence relation]], let {{M|(X,\mathcal{J})}} be a [[topological space]]. We get a map, {{M|\pi:X\rightarrow\frac{X}{\sim} }} that takes {{M|\pi:x\rightarrow[x]}} | ||
* The quotient topology on {{M|\frac{X}{\sim} }} is the [[finer topology|finest]] such that {{M|\pi}} is [[continuous]] | * The quotient topology on {{M|\frac{X}{\sim} }} is the [[finer topology|finest]] such that {{M|\pi}} is [[continuous]] | ||
| Line 14: | Line 19: | ||
* {{M|1=x\sim y\iff q(x)=q(y)}} and that's where this comes from | * {{M|1=x\sim y\iff q(x)=q(y)}} and that's where this comes from | ||
===Passing to the quotient=== | ===Passing to the quotient=== | ||
| − | This is very similar to [[quotient (function)|the quotient of a function]].<br/> | + | <div style="float:right;margin:0.1em;"> |
| − | *Let X and Z be [[topological space|topological spaces]] | + | {| class="wikitable" border="1" |
| + | |- | ||
| + | |<center><span style="font-size:1.2em;"><m>\xymatrix{ | ||
| + | X \ar[d]_q \ar[dr]^f & \\ | ||
| + | Y \ar@{.>}[r]_{\bar{f}} & Z | ||
| + | }</m></span></center> | ||
| + | |- | ||
| + | ! Passing to the quotient | ||
| + | |} | ||
| + | </div>This is very similar to [[quotient (function)|the quotient of a function]].<br/> | ||
| + | * Let X and Z be [[topological space|topological space|topological spaces]], | ||
| + | * let {{M|q:X\rightarrow Y}} be a quotient map, | ||
| + | * let {{M|f:X\rightarrow Z}} be ''any'' continuous mapping such that {{M|1=q(x)=q(y)\implies f(x)=f(y)}} | ||
| + | Then | ||
| + | * There exists a unique continuous map, {{M|\bar{f}:Y\rightarrow Z}} such that {{M|1=f=\bar{f}\circ q}} | ||
| + | |||
==Munkres== | ==Munkres== | ||
| + | '''Munkres starts with a quotient map''' | ||
| + | * Let {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} be [[topological space|topological spaces]] and | ||
| + | * let {{M|q:X\rightarrow Y}} be a [[surjective]] map | ||
| + | We say {{M|q}} is a ''quotient map'' provided: | ||
| + | * {{M|\forall U\in\mathcal{P}(Y)[U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}]}} | ||
| + | He goes on to say: | ||
| + | # This condition is "stronger than continuity" (of {{M|q}} presumably) probably because if we gave {{M|Y}} the [[indiscrete topology]] it'd be continuous. | ||
| + | # He defines this in several ways, one of which is "saturation" of maps. Yeah this is just the [[equivalence relation]] version hiding (CHECK THIS THOUGH) | ||
| + | ===Quotient topology=== | ||
| + | If {{M|(X,\mathcal{J})}} is a [[topological space]] and {{M|A}} a [[set]] and if {{M|p:X\rightarrow A}} is a ''[[surjective]]'' [[map]] then: | ||
| + | * There is exactly on topology, {{M|\mathcal{K} }} on {{M|A}} relative to which {{M|p}} is a ''quotient map'' (as defined above) | ||
| + | That topology is the ''quotient topology'' induced by {{M|p}} | ||
| + | ===Quotient space=== | ||
| + | Let {{M|(X,\mathcal{J})}} be a [[topological space]] and let {{M|X^*}} be a [[partition]] of {{M|X}} into disjoint subsets whose union is {{M|X}} (that is the definition of a [[partition]]). | ||
| + | Let {{M|p:X\rightarrow X^*}} be the surjective map that carries each point of {{M|X}} to the element of {{M|X^*}} containing that point, then: | ||
| + | * The ''quotient topology'' induced by {{M|p}} on {{M|X^*}} is called the ''quotient space'' of {{M|X}} | ||
| + | ==Mond== | ||
| + | : '''Note: ''' David Mond is my tutor at university. While I ''do not'' like his style of writing (informal definitions, ambiguities in the ''English'' interpretation) he does have a great way of ordering things. That is applicable here. I also found like 9 typos in the first 8 pages; however it has many examples and many pictures, and they are lecture notes. | ||
| + | Mond starts with a [[topological space]] {{M|(X,\mathcal{J})}} and an [[equivalence relation]], {{M|\sim}}. Then: | ||
| + | * The ''quotient topology'' ({{M|\mathcal{K} }}) on {{M|\frac{X}{\sim} }} is the topology where {{M|\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}]}} | ||
| + | ===Passing to the quotient=== | ||
| + | Mond then goes for passing to the quotient, exactly as John M. Lee has. Very weirdly worded though. | ||
| + | |||
| + | No mention of the ''quotient map'' as a concept. | ||
| + | |||
| + | Deals with [[equivalence relation generated by]] which is great. | ||
| + | ==Mendelson== | ||
| + | Starts with what is called a ''quotient map'' above, but calls it "an identification". Then he goes straight on to "passing to the quotient" it's a very weirdly written section, but he does: | ||
| + | # Identification map | ||
| + | # Identification topology | ||
| + | It seems I'll have to prove these concepts are the same (having one induces the other). Provides a source for the "identification" terminology, which is useful. | ||
| + | ==Gamelin & Greene== | ||
| + | # Quotient topology, given a [[topological space]] {{M|(X,\mathcal{J})}} and a set {{M|Y}}, as usual. | ||
| + | # Slightly strange, again top space {{M|(X,\mathcal{J})}} and an equivalence relation {{M|\sim}}, for {{M|f:\frac{X}{\sim}\rightarrow Y}} - {{M|f}} is continuous {{M|\iff}} {{M|f\circ\pi}} is continuous ({{M|\pi}} being the canonical projection) | ||
| + | #* This is NOT passing to the quotient | ||
| + | # Passing to the quotient now. Let {{M|f:X\rightarrow Y}} be continuous function, and {{M|\sim}} and equivalence relation on {{M|X}} such that {{M|f}} is constant for each {{M|x\in[x]\in\frac{X}{\sim} }}. Then: | ||
| + | #* {{M|\exists g:\frac{X}{\sim}\rightarrow Y}} continuous such that {{M|1=f=g\circ\pi}} | ||
Latest revision as of 19:45, 26 April 2016
- Note to readers: the page quotient topology as it stands right now (Alec (talk) 17:07, 21 April 2016 (UTC)) is an embarrassment to me. However before I can clean it up I must unify it. I've been using it for almost 2 years now though I promise! Gosh this is embarrassing.
See Notes:Quotient topology plan for an outline of the page.
Contents
According to John M. Lee
- Addendum: for some reason I lie here, the author considers a map [ilmath]f:X\rightarrow A[/ilmath] where [ilmath]A[/ilmath] is a set and later applies the equivalence relation version to it.
Let [ilmath]\sim[/ilmath] denote an equivalence relation, let [ilmath](X,\mathcal{J})[/ilmath] be a topological space. We get a map, [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath] that takes [ilmath]\pi:x\rightarrow[x][/ilmath]
- The quotient topology on [ilmath]\frac{X}{\sim} [/ilmath] is the finest such that [ilmath]\pi[/ilmath] is continuous
Let [ilmath]\mathcal{K} [/ilmath] denote a topology on [ilmath]\frac{X}{\sim} [/ilmath], then we may define [ilmath]\mathcal{K} [/ilmath] as:
- [ilmath]\mathcal{K}:=\{U\in\mathcal{P}(\frac{X}{\sim})\ \vert\ \pi^{-1}(U)\in\mathcal{J} \}[/ilmath], that is:
- [ilmath]U\in\mathcal{P}(\frac{X}{\sim})[/ilmath] is open if [ilmath]\pi^{-1}(U)[/ilmath] is open in [ilmath]X[/ilmath] - we get "only if" by going the other way. I must make a page about how definitions are "iff"s
Note: more than one book is very clear on "[ilmath]U\in\mathcal{P}(\frac{X}{\sim})[/ilmath] is open in [ilmath]\frac{X}{\sim} [/ilmath] if and only if [ilmath]\pi^{-1}(U)\in\mathcal{J} [/ilmath], not sure why they stress it so.
Quotient map
A map between two topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] is a quotient map if:
- It is surjective
- The topology on [ilmath]Y[/ilmath] ([ilmath]\mathcal{K} [/ilmath]) is the quotient topology that'd be induced on [ilmath]Y[/ilmath] by the map [ilmath]q[/ilmath]
Lee actually defines the quotient topology using maps first, then constructs the equiv relation version, but we can can define an equivalence relation as follows:
- [ilmath]x\sim y\iff q(x)=q(y)[/ilmath] and that's where this comes from
Passing to the quotient
| Passing to the quotient |
|---|
- Let X and Z be topological space|topological spaces,
- let [ilmath]q:X\rightarrow Y[/ilmath] be a quotient map,
- let [ilmath]f:X\rightarrow Z[/ilmath] be any continuous mapping such that [ilmath]q(x)=q(y)\implies f(x)=f(y)[/ilmath]
Then
- There exists a unique continuous map, [ilmath]\bar{f}:Y\rightarrow Z[/ilmath] such that [ilmath]f=\bar{f}\circ q[/ilmath]
Munkres
Munkres starts with a quotient map
- Let [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] be topological spaces and
- let [ilmath]q:X\rightarrow Y[/ilmath] be a surjective map
We say [ilmath]q[/ilmath] is a quotient map provided:
- [ilmath]\forall U\in\mathcal{P}(Y)[U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}][/ilmath]
He goes on to say:
- This condition is "stronger than continuity" (of [ilmath]q[/ilmath] presumably) probably because if we gave [ilmath]Y[/ilmath] the indiscrete topology it'd be continuous.
- He defines this in several ways, one of which is "saturation" of maps. Yeah this is just the equivalence relation version hiding (CHECK THIS THOUGH)
Quotient topology
If [ilmath](X,\mathcal{J})[/ilmath] is a topological space and [ilmath]A[/ilmath] a set and if [ilmath]p:X\rightarrow A[/ilmath] is a surjective map then:
- There is exactly on topology, [ilmath]\mathcal{K} [/ilmath] on [ilmath]A[/ilmath] relative to which [ilmath]p[/ilmath] is a quotient map (as defined above)
That topology is the quotient topology induced by [ilmath]p[/ilmath]
Quotient space
Let [ilmath](X,\mathcal{J})[/ilmath] be a topological space and let [ilmath]X^*[/ilmath] be a partition of [ilmath]X[/ilmath] into disjoint subsets whose union is [ilmath]X[/ilmath] (that is the definition of a partition). Let [ilmath]p:X\rightarrow X^*[/ilmath] be the surjective map that carries each point of [ilmath]X[/ilmath] to the element of [ilmath]X^*[/ilmath] containing that point, then:
- The quotient topology induced by [ilmath]p[/ilmath] on [ilmath]X^*[/ilmath] is called the quotient space of [ilmath]X[/ilmath]
Mond
- Note: David Mond is my tutor at university. While I do not like his style of writing (informal definitions, ambiguities in the English interpretation) he does have a great way of ordering things. That is applicable here. I also found like 9 typos in the first 8 pages; however it has many examples and many pictures, and they are lecture notes.
Mond starts with a topological space [ilmath](X,\mathcal{J})[/ilmath] and an equivalence relation, [ilmath]\sim[/ilmath]. Then:
- The quotient topology ([ilmath]\mathcal{K} [/ilmath]) on [ilmath]\frac{X}{\sim} [/ilmath] is the topology where [ilmath]\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}][/ilmath]
Passing to the quotient
Mond then goes for passing to the quotient, exactly as John M. Lee has. Very weirdly worded though.
No mention of the quotient map as a concept.
Deals with equivalence relation generated by which is great.
Mendelson
Starts with what is called a quotient map above, but calls it "an identification". Then he goes straight on to "passing to the quotient" it's a very weirdly written section, but he does:
- Identification map
- Identification topology
It seems I'll have to prove these concepts are the same (having one induces the other). Provides a source for the "identification" terminology, which is useful.
Gamelin & Greene
- Quotient topology, given a topological space [ilmath](X,\mathcal{J})[/ilmath] and a set [ilmath]Y[/ilmath], as usual.
- Slightly strange, again top space [ilmath](X,\mathcal{J})[/ilmath] and an equivalence relation [ilmath]\sim[/ilmath], for [ilmath]f:\frac{X}{\sim}\rightarrow Y[/ilmath] - [ilmath]f[/ilmath] is continuous [ilmath]\iff[/ilmath] [ilmath]f\circ\pi[/ilmath] is continuous ([ilmath]\pi[/ilmath] being the canonical projection)
- This is NOT passing to the quotient
- Passing to the quotient now. Let [ilmath]f:X\rightarrow Y[/ilmath] be continuous function, and [ilmath]\sim[/ilmath] and equivalence relation on [ilmath]X[/ilmath] such that [ilmath]f[/ilmath] is constant for each [ilmath]x\in[x]\in\frac{X}{\sim} [/ilmath]. Then:
- [ilmath]\exists g:\frac{X}{\sim}\rightarrow Y[/ilmath] continuous such that [ilmath]f=g\circ\pi[/ilmath]
