What is a $1$-point space? At the beginning of chapter 3 of Gilbert Strang's Introduction to Linear Algebra it says: "The "vectors" in $S$ can be matrices or functions of $x$. The $1$-point space $Z$ consists of $x=0$"
What is a $1$-point space?
Do we have $2$-point, $3$-point spaces? what are they?
 A: A vector space over an infinite field $\Bbb K$ (like $\Bbb R$ or $\Bbb C$) has either one or infinitely many points. The one-point space (as it is called in your book) seems to be the set $Z=\{0\}$ only containing the zero-vector (vector with no direction and of length zero) and nothing else.
The thing with a vector space $V$ is that for any two points $x,y\in V$ (or call them vectors if you want), all the points "between" $x$ and $y$ must also belong to $V$ (among others). And these are infinitely many (at leats for $x\not=y$).
On the other hand, if you have only a single point/vector $0$, there are no two different points for which we can ask the "between-question". So one point is okay. Two are not. Infinitely many are okay again if arranged appropriately.
The story is a bit more complex for finite fields $\Bbb K=\Bbb F_q$.

From a geometric point of view (assuming $\Bbb K=\Bbb R$) you can think of $0,1,2,3,...$ dimensional spaces as


*

*$0$-dimensional: a single point

*$1$-dimensional: a single infinite line

*$2$-dimensional: a single infinite plane

*$3$-dimensional: all of $3$D space

*...


All of them are connected, i.e. you can reach any point from any other point by traveling a "straight" path. This is not possible in a space only consisting of two points (or any other finite amount $>1$). The "in-between points" are missing (if you already know something about linear combinations: I am talking about points $\alpha x+\beta y$ for $\alpha,\beta\in\Bbb R$).
A: Notice that when you are dealing with a field of characteristic zero such as the real numbers, if you have two points (vectors) $\vec{x_1}$ and $\vec{x_2}$, then all points (vectors) $n(\vec{x_1}-\vec{x_2})$ for any $n \in \mathbb{N}$ will be different. So, the answer is no in this case. One you have more than a single point, you will get infinitely many points.
Can you have a vector field of $2$ points working over a field of a different characteristic? 
As an example, if you work with $\mathbb{Z}_2$ over itself as a vector space, then the answer is positive and it has two points. But people mainly like to work over fields of characteristic zero.
A: I don't have the book in front of me, but I suppose that a “$1$-point space” is a vector space with a single point. There is one and only one such space: $\{0\}$. Unless you are working over finite fields, there are no vector spaces with exactly $2$ or $3$ elements. In fact, other than $\{0\}$ , they all have infinitely many elements.
