Undergrad Homemorphism in quotient topology

[elias001]

Questions

I have two general questions about the topic of quotient topology.. Suppose I have a set $X$ and I defined an equivalence relation $\sim$ on $X$ and I want to know what quotient toplogical sapce is hoemeorphic to. I have included a list of definitions, lemmas, propositions and theorems from two different textbooks in the Background section below. Specifically the two question I have is one,

$Question 1$ If given a set $X$ and an equivalence relation $\sim$ on $X$, sometimes an exercise in a topology text would ask the reader to identify what the quotient space is or what the equivalence relation defined on set $X$ is homeomorphic to. How does one usually go about such exercise. I assumed by visualizing or the process of identifying, it requires one to provide a proof.

$Question 2$ Suppose in $Question 1$, I know what the quotient topological space is but the construction can be only describe using picture/verbal descriptive type style proofs. To be specific, take the mobius strip, everyone has seen how they have seen a rectangular strip $R$ with direction arrows running along each of the four edges, and we have the equivalence relations $(x,0)\sim (1-x,1)$, one is suppose to identify the opposite edges of the rectangular strip $R$ by doing a $180$ degree twist. Depending on the sophiscation of the textbook, a reader might be presented an parametric representation of the mobius strip, basically what one would see in a differential topology text,, in terms of charts. This last two points, are important because depending on the type of quotient topological space, a parametric representation of it might not always be readily know. So how does one go about showing homeomorphism with just a given equivalence relation.


Background

The following are from Topology by Murray Eisenberg and Elements of Algebraic Topology by: Anat Shastri

[Topology by Murray Eisenberg]

Lemma 1 Let $\sim$ be an equivalence relation on a topological space $X$, and let $p:X\to X/\sim$ be the quotient map. Then the collection
$$T=\{V\subset X/\sim:p^{-1}(V)\text{ open in }X\}$$

is a topology on the quotient set $X/\sim.$

Definition 1 The quotient of a topological space $X$ under an equivalence relation$\sim$ is the topological space whose underlying set is $X/\sim$ and whose topology is the collection $T$ described above. This topology is called the quotient topology.

Proposition 1: Let $p:X\to X/sim$ be the quotient map induced by an equivalence relation $\sim$ on a topological space $X.$ Then the quotient topology is the greatest topology on $X/\sim$ making $p$ continuous.


Theorem 1: A map

$$g:X/\sim\to Y$$

from a quotient space $X/\sim$ into a topological space $Y$ is continuous if and only if its composite

$$g\circ p:X\to Y$$

with the quotient map $p:X\to X/\sim$ is continuous..

Theorem 2: Let $p:X\to X/\sim$ be the quotient map induced by an equivalence relation $\sim$ on a topological space $X.$ Let $f:X\to Y$ be a continuous map from $X$ into a topological space $Y$ that is constant on each equivalence class under $\sim,$ that is,

$$x\sim t\quad \Rightarrow\quad f(x)=f(y)\quad\quad (x,t,\in X).$$

Then there is a unique continuous map

$$f^{*}:X/\sim\to Y$$

such that

$$f^{*}\circ p=f.$$

Moreover:
$\quad (1)$ The map $f^{*}$ is surjective if $f$ is surjective.
$\quad (2)$ The map $f^{*}$ is injective if $f$ takes distinct values at representatives of different equivalence classes under $\sim.$
$\quad (3)$ The map $f^{*}$ is open if $f$ is injective. and the open subsets of $Y$ are those subsets $W$ of $Y$ for which $f^{-1}(W)$ is open in $X.$

Proposition 2: Each continuous open surjection and each continuous closed surjection is a quotient map.

[Elements of Algebraic Topology by: Anat Shastri]

Let $q:(X,\tau)\to (Y,\tau')$ be a surjective map (i.e. continuous function) of topological spaces.

Lemma 2: The following statements are equivalent.

$(i)$ $U\in \tau'$ iff $q^{-1}(U)\in\tau.$
$(ii)$ A function $g:(Y,\tau')\to (Z,\tau'')$ is continuous iff $g\circ q$ is continuous.
$(iii)$ For a fixed $\tau,\tau'$ is the maximal topology on $Y$ such that $q$ is continuous.

Definition 2: Under the above conditions, we say $(Y,\tau')$ is a quotient space of $(X,\tau)$ and the map $q$ is called a quotient map.




Thank you in advance

 

[pasmith]

It should not be difficult to get from a graphical description to a parametrisation.

For example, for the Mobius strip one can take a line segment with centre on a circle of radius R such that the angle to the horizontal plane goes through a half rotation as the centre moves through a full rotation. That leads to something like
\begin{split}x &= (R + (u-\tfrac12)\cos (\pi v))\cos (2\pi v) \\y &= (R + (u-\tfrac12) \cos (\pi v))\sin (2\pi v) \\z &= R + (u-\tfrac12) \sin (\pi v). \end{split}
for R > 1.

 

[elias001]

@pasmith Is it always necessary to prove homeomorphism by finding explicit parametrization from graphical descripption, or can one do it by relying solely on the description based on equivalence relations?

 

[pasmith]

A graphical description is not really a rigorous proof, although it might help you to find one.

Ultimately, showing that X/\sim is homeomorphic to Y requires finding a continuous function f: X/\sim \to Y and showing that it has a continuous inverse.

For example, showing that [0,1]/\sim where 0 \sim 1 and otherwise x \sim y \Leftrightarrow x = y is homeomorphic to S^1 can be done via the map [x] \mapsto (\cos (2\pi x), \sin (2\pi x)).

 

[elias001]

@pasmith what about the case of the klein bottle or roman surface? For the klein bottle, I only find how is parametrize in $R^3$ in two different books about differential forms.

 

[jbergman]

Questions

I have two general questions about the topic of quotient topology.. Suppose I have a set $X$ and I defined an equivalence relation $\sim$ on $X$ and I want to know what quotient toplogical sapce is hoemeorphic to. I have included a list of definitions, lemmas, propositions and theorems from two different textbooks in the Background section below. Specifically the two question I have is one,

$Question 1$ If given a set $X$ and an equivalence relation $\sim$ on $X$, sometimes an exercise in a topology text would ask the reader to identify what the quotient space is or what the equivalence relation defined on set $X$ is homeomorphic to. How does one usually go about such exercise. I assumed by visualizing or the process of identifying, it requires one to provide a proof.

$Question 2$ Suppose in $Question 1$, I know what the quotient topological space is but the construction can be only describe using picture/verbal descriptive type style proofs. To be specific, take the mobius strip, everyone has seen how they have seen a rectangular strip $R$ with direction arrows running along each of the four edges, and we have the equivalence relations $(x,0)\sim (1-x,1)$, one is suppose to identify the opposite edges of the rectangular strip $R$ by doing a $180$ degree twist. Depending on the sophiscation of the textbook, a reader might be presented an parametric representation of the mobius strip, basically what one would see in a differential topology text,, in terms of charts. This last two points, are important because depending on the type of quotient topological space, a parametric representation of it might not always be readily know. So how does one go about showing homeomorphism with just a given equivalence relation.


Background

The following are from Topology by Murray Eisenberg and Elements of Algebraic Topology by: Anat Shastri

[Topology by Murray Eisenberg]

Lemma 1 Let $\sim$ be an equivalence relation on a topological space $X$, and let $p:X\to X/\sim$ be the quotient map. Then the collection
$$T=\{V\subset X/\sim:p^{-1}(V)\text{ open in }X\}$$

is a topology on the quotient set $X/\sim.$

Definition 1 The quotient of a topological space $X$ under an equivalence relation$\sim$ is the topological space whose underlying set is $X/\sim$ and whose topology is the collection $T$ described above. This topology is called the quotient topology.

Proposition 1: Let $p:X\to X/sim$ be the quotient map induced by an equivalence relation $\sim$ on a topological space $X.$ Then the quotient topology is the greatest topology on $X/\sim$ making $p$ continuous.


Theorem 1: A map

$$g:X/\sim\to Y$$

from a quotient space $X/\sim$ into a topological space $Y$ is continuous if and only if its composite

$$g\circ p:X\to Y$$

with the quotient map $p:X\to X/\sim$ is continuous..

Theorem 2: Let $p:X\to X/\sim$ be the quotient map induced by an equivalence relation $\sim$ on a topological space $X.$ Let $f:X\to Y$ be a continuous map from $X$ into a topological space $Y$ that is constant on each equivalence class under $\sim,$ that is,

$$x\sim t\quad \Rightarrow\quad f(x)=f(y)\quad\quad (x,t,\in X).$$

Then there is a unique continuous map

$$f^{*}:X/\sim\to Y$$

such that

$$f^{*}\circ p=f.$$

Moreover:
$\quad (1)$ The map $f^{*}$ is surjective if $f$ is surjective.
$\quad (2)$ The map $f^{*}$ is injective if $f$ takes distinct values at representatives of different equivalence classes under $\sim.$
$\quad (3)$ The map $f^{*}$ is open if $f$ is injective. and the open subsets of $Y$ are those subsets $W$ of $Y$ for which $f^{-1}(W)$ is open in $X.$

Proposition 2: Each continuous open surjection and each continuous closed surjection is a quotient map.

[Elements of Algebraic Topology by: Anat Shastri]

Let $q:(X,\tau)\to (Y,\tau')$ be a surjective map (i.e. continuous function) of topological spaces.

Lemma 2: The following statements are equivalent.

$(i)$ $U\in \tau'$ iff $q^{-1}(U)\in\tau.$
$(ii)$ A function $g:(Y,\tau')\to (Z,\tau'')$ is continuous iff $g\circ q$ is continuous.
$(iii)$ For a fixed $\tau,\tau'$ is the maximal topology on $Y$ such that $q$ is continuous.

Definition 2: Under the above conditions, we say $(Y,\tau')$ is a quotient space of $(X,\tau)$ and the map $q$ is called a quotient map.




Thank you in advance

I don't think this is an easy question in general. Yhere is no magic answer. Every smooth manifold can be described as a quotient of patches of $\mathbb R^n$. If this problem was mechanical then problems like the Poincare conjecture would be simple.

Now if you just want to embed the quotient space in $\mathbb R^{2n}$ then this is an easier question at least in the smooth category via the Whitney Embedding theorem.