Is there anything special about the below finite metric space? See below for details. I am a high school student who has been playing around with certain mathematical ideas, most recently metric spaces, and I believe I have just "defined" if you will, the following metric space:
Metric space: $(X,d)$
Set: $X={(a,a),(a,b),(b,a),(b,b)}$
Metric:
$$
d((x_1,x_2),(y_1,y_2)) =\left\{ \begin{array}[ll]\\
0 & \text{if } x_1 = y_1 \text{ and } x_2 = y_2.\\
1 & \text{if } x_1 = y_1 \text{ xor } x_2 = y_2.\\
\sqrt{2} & \text{if } x_1 \ne y_1 \text{ and } x_2 \ne y_2.
\end{array}\right.
$$ This seems really interesting to me as it seems to describe the lengths of the sides and diagonals of a square. My questions are these, in no particular order:
1: Is this really a metric space? $d$ seems to fit the triangle inequality, but perhaps I'm missing something.
2: If this is a metric space, does it have any interesting properties? Also, are there any useful metric spaces like this (have any been used to prove/disprove conjectures? 
3: Is this similar to any other idea/concept/example in metric spaces that I could look at to gain a deeper appreciation?
4: On a more general note, are there any good introductory texts to metric spaces, that don't require much in the way of current knowledge. I do know some set theory, and can read (some) proofs, if that helps.
Please remember I am only a novice in this area, so please do not assume any background knowledge, outside of the definition of a metric space.
 A: *

*Yes, this is a metric space. It is (isomorphic to) the subspace of $\mathbb R^2$ consisting of the four points $(0, 0), (0, 1), (1, 0), (1, 1)$, with the metric inherited from the metric on $\mathbb R^2$.


2,3. Nothing particular that I can think of.


*Most texts for a first class in real analysis should give an overview of metric space theory. I personally like Charles Pugh's Real Mathematical Analysis quite a bit, but any such text will do.

A: 1) To test the triangle inequality there are only the following  situations with $X = (x_1,x_2), Y=(y_1,y_2), Z=(z_1,z_2)$
a.) X = Y.  Then $d(X,Y) + d(Y,Z) = 0 + d(X,Z) = d(X,Z) \le d(X,Y) + d(Y,Z)$.  It holds.
Symmetry assures it holds for $Y = Z$.
b) Otherwise $X \ne Y$ and $Y \ne Z$.  
$d(X,Z) = \{0,1,$ or $\sqrt{2}\}$ so $d(X,Y) \le \sqrt{2}$; $d(X,Y) = \{1,$ or $\sqrt{2}\}$; $d(Y,Z) = \{1,$ or $\sqrt{2}\}$ so $d(X,Y) + d(Y,Z) = \{2, 1+\sqrt{2}, 2\sqrt{2}\}$. so $2 \le d(X,Y) + d(Y,Z)$
So $d(X,Z) \le \sqrt{2} < 2 \le d(X,Y) + d(Y,Z)$ so triangle inequality holds.
The other conditions of the metric space are obviously true.  ($d(X,Y) \in \mathbb R$; $d(X,Y) \ge 0$; $d(X,Y) = d(Y,X)$; $d(X,Y) = 0 \iff X = Y$).
2) Notice it isn't just a metric space on your set X.  It's a metric space on all $\mathbb R^2$.  Or on any cross product set $S \times S$.
I don't know if you know anything about topology of metric spaces but this makes all sets both open and closed.  (I'll explain that further if you like.)
3) Look at the discrete/trivial metric space which is the simplest most basic metric (on any set).  $d(x,y) = 1 \iff x \ne y; d(x,y) = 0 \iff x = y$.  It's very similar to yours but simpler.
Consider the paris railway metric:  Set: $X$ with a central element $c$.  If $x = y$ $d(x,y) = 0$.  Otherwise $d(c,x) = r_x$ where $r_x$ can be an arbitrary positive real number of each element $x$.  And if $x \ne c$ and $y \ne c$ then $d(x,y) = d(c,x) + d(c,y)$.  This is called a the paris railway metric becuase to get from any two points you must pass through the "main terminal" $c$.
It's not actually that interesting a metric space but I like it because it (and the discrete metric) shows us all the properties required of a metric while getting us away from our preconceptions of a "distance" being "as the crow flies".
Another metric space is Set: $\mathbb R^2$ and the metric is $d_2(X,Y) = \frac{d_1(X,Y)}{d_1(X,Y) + 1}$ where $d_1(X,Y)$ is the usual euclidean metric: $d_1((x_1,y_1),(x_2,y_2)) = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2)}$.
Whats interesting about this metric is that it maintains nearly on the other properties of $R^2$ with euclidean metric and "shapes" in euclidean $R^2$ will be the equivalent "shapes" in this metric space to.  But this metric space is bounded.  All distances are less than 1 so there are no infinite distances and all sets are bounded.
Also to $d_2 = \frac{d_1}{d_1 + 1}$ can be applied to any metric space to make an equivalent but bounded new space.
4)  Hmmmm.... not off the top of my head.
