Questions concerning quotient spaces specifically collapsing a set to a point. I am trying to get some practice in picturing quotient spaces and have found the list of exercises from a topology text. (Introduction to Topology Pure and Applied by Colin Adams and Robert Franzosa)  I have specific questions about them and I am hoping if the community can provide me with some clarifications and also if my attempted description of each quotient spaces are accurate.  All of them has concerns the concept of collapsing a set to a point.  My apologises in advance if I did not attached any kind of visual illustrations..
$1)~$ The interval $[0,4]$ as a subspace of $\mathbb{R}$, with integer points identified with each other.
Question:  The subspace would be $\{0, 1, 2, 3, 4\}=[0,4]\cap\mathbb{R}$, would equivalence classes would consists of these type of following subsets:  $~\{(a, b, c, d, e), (a,b), (a,b,c), (a, b, c, d)\}~$
$2)~$ The interval $[0,9]$ as a subspace of $\mathbb{R}$, with even integer points identified with each other to form a point and with odd integer points identified with each other to form a different point.
Question:  This would be similar to the previous question, except we have two different sets of equivalence classes.  
$a)~$ $[2x]=\{0, 2, 4, 6, 8\}=[0,9]\cap\mathbb{R}$
$b)~$ $[2x+1]=\{1, 3, 5, 7, 9\}=[0,9]\cap\mathbb{R}$
Would the visual description be real number line $\mathbb{R}$ with two dots one label $2x$ and the other label $2x+1$
$3)~$ The real line $\mathbb{R}$ with $[-1,1]$ collapsed to a point.
Question:  Here when a closed interval is collapsed to a point, does it mean 
the following $x\text{~}y$ iff $x=y$ for all $x$, and $y$ in $[a,b]$.  Hence $a=b=x$ for all $x\in [a,b]$  The thing is, when it states collapsed to a point, do just let the point be an arbitrary point $y$ outside the closed interval.  The the number line $\mathbb{R}$ would have that closed interval erased. Is that a correct visual description? 
$4)~$ The real line $\mathbb{R}$ with $(-1,1)$ collapsed to a point.
Question:  Is the equivalence relation $x\text{~}y$ iff $x=y$ where $x,y\in (-1.1)$ and both $x$, $y$ do not equal to $-1$ and $1$.  So the description of the number line $\mathbb{R}$ is an open dot on the number line where the open dot equal to both $-1$ and $1$ and any elements within $(-1,1)$ disappears.

$5)~$ The real line $\mathbb{R}$ with $(-1,1]$ collapsed to a point, similarly for the case $[-1.1)$
Question:  Since this is a semi open interval, if elements of the interval $(-1,1]$ are all consider to be equal, with the exception of $-1$.  Then would the number line $\mathbb{R}$ be that there would be an open dot on the number line and the open dot is at $-1$. 
$6)~$ The real line $\mathbb{R}$ with $[-2,-1]~\cup~ [1,2]$ collapsed to a point. 

Question:  Same reasoning are applied to each respective closed intervals $[-2,-1]$ and $[1,2]$ from $(3)$ and the number line would consist of $2$ closed dots separated by a distance of $2$ integer units, 
$7)~$ The plane $\mathbb{R}^2$ with the circle $S^1$ collapsed to a point.
Question:  Is the description consist of an arbitrary point $x$ of the circle, be $x$ at its boundary or at the interior of the circle, every other element of the circle both from its boundary and its interior shrinks to the point $x$
$8)~$ The plane $\mathbb{R}^2$ with the circle $S^1$ and the origin collapsed to a point.
Question:  Here the circle at the origin, an arbitrary element $z$ of the circle minus $(0,0)$,  all the elements of the circle's boundary and its interior shrinks to point $z$
$9)~$ The sphere with the north and south pole identified with each other.
Question:  Would the visual description be two spheres tangent to each other at a point, since only $(0,0,1)\text{~}(0,0,-1)$ iff $(0,0,1)=(0,0,-1)$
$10)~$ The sphere with the equator collapsed to a point.
Question:  I am guessing the equivalence relation is as follows: $(\cos(\theta_1), \sin(\theta_1), 0)\text{~}(\cos(\theta_2), \sin(\theta_2), 0)$ iff $\theta_1\text{~}\theta_2$ and $\theta_1,\theta_2 \in [0, 2\pi]$ and the picture is the joining of two spheres both of same radius at a point of tangency, one on top of another.
Thank you in advance
 A: I'm afraid you're not interpreting these correctly. I'm not going to go over all of them (after all, you've asked ten questions here!), but let's look at least at the first one.
(1) I don't understand what you mean when you say

… the subspace would be $\{0,1,2,3,4\}=[0,4]\cap\mathbb{R}$ …

In this question, $[0,4]$ is the given topological subspace of the topological space $\mathbb{R}$. By implementing the given identification, we will get a new topological space (which, by the way, won't be a subspace of $\mathbb{R}$ anymore).
By definition of quotient topology, the quotient space consists of all equivalence classes. In this example, since we're only told to identify all integer points of $[0,4]$ with each other, $\{0,1,2,3,4\}$ is indeed something important — it's a new equivalence class, which becomes a single point of the new topological quotient space. We can call it $[0]=\{0,1,2,3,4\}$, for example. Note that since we're not identifying anything else, all other equivalence classes are, informally speaking, just the original points. For example, $[0.5]=\{0.5\}$, $[\pi]=\{\pi\}$, etc. So the new topological space has the following equivalence classes as its elements:


*

*$[0]=\{0,1,2,3,4\}$ as one of its points;

*$[x]=\{x\}$ for each $x\in(0,1)\cup(1,2)\cup(2,3)\cup(3,4)$.
Geometrically, imagine the following. Take a piece of thread to represent the segment $[0,4]$; mark the points $0,1,2,3,4$ on it, $0$ and $4$ being the endpoints; and then glue together $0,1,2,3,4$ to get a four-petal flower shape — where each of the four petals represents one of the intervals $(0,1)$, $(1,2)$, $(2,3)$, and $(3,4)$, and the center is this "new" point $[0]=\{0,1,2,3,4\}$:

I hope this explanation is going to help you with the other exercises.
