Counter example that a convex set is not necessarily an interval or ray . The example I am trying to frame is like this :

I am considering the dictionary order on $\mathbb{R}^2$ , now I know that the set $S=\{x^2+y^2<1\}$ is a convex set(I am confused that it is with respect to which order).It doesn't look like that $S$ is an interval or ray.

I need some help to frame this question properly to give an example.
 A: From your recent posts, I think the multiple usages of "convex" in different contexts is causing confusion. Hopefully the following sheds some light.

Definition 1. (Convex, in the context of vector spaces)
A subset $S \subset \Bbb R^n$ is said to be convex if for all $x, y \in S$ and $t \in [0, 1]$, it is the case that $tx + (1 - t)y \in S$.
(In general, you can replace $\Bbb R^n$ with any $\Bbb R$-vector space $V$, if you know what those are.)
The above definition says that given two points in $S$, the line segment joining the two points is contained in $S$.

Definition 2. (Convex, as in Munkres)
Let $L$ be a simply ordered set. $S \subset L$ is convex in $L$ if given any $a, b \in S$ and $c \in L$ with $a < c < b$, it is the case that $c \in S$.
In other words, given two points in $S$ and any point of $L$ in between them, that point must actually be in $S$. (Note that this depends on your bigger set $L$.)
The above is actually sometimes the definition of an interval. But Munkres defines that in another manner.

Now, the two definitions above coincide for the case that $L = \Bbb R$ but for $\Bbb R^n$ (with $n > 1$), it does not. Firstly, to apply the second definition to $\Bbb R^n$, we would need an order on it. The usual choice would be the dictionary order but the definitions of convex won't coincide.
The $S$ described in your post is an example of this. (Do you see how?)

Definition 3. (Interval, as in Munkres)
Let $L$ be a simply ordered set. An interval is a set which is of one of the following forms:

*

*$(a, b)$,

*$[a, b)$,

*$(a, b]$,

*$[a, b]$,

for some $a, b \in L$.
The definitions of those sets are the usual one.

It is easy to see that all intervals are convex but the converse is not true.
Consider the simply ordered set $L = \Bbb Q$ in the usual order and the subset $$S = \{x \in \Bbb Q : x^2 < 2\} \subset \Bbb Q.$$
Then, $S$ is convex (in $\Bbb Q$!) but it not of the above four forms.
Munkres also defines what a ray is. I leave it to you to verify that $S$ is not a ray either.

Moral. The same words have different meaning in different parts of math and you should be careful in seeing what they are defined as.
