for $I = [0,1]$, is $I\times I$ convex in $\mathbb{R} \times \mathbb{R}$? for $I = [0,1]$, is $I \times I$ convex in $\mathbb{R} \times \mathbb{R}$?

The definition of convex seems to be that


  $Y \subset X$ is convex in $X$ if $\forall a &lt; b $ in $Y$ whole of $(a,b)$ in $X$ lies in  $Y$. 


For example, if I take $(1/3, 1/3) &lt; (2/3,1/3)$ in $I \times I $ then, the rectangular strip in $\mathbb{R} \times \mathbb{R}$ $( (1/3, 1/3) , (2/3,1/3)$ is obviously not entirely in  $I \times I$. 

So, this subset $I \times I$ of $\mathbb{R} \times \mathbb{R}$ is not convex. 

Am I doing it correctly? I will really appreciate some help on the matter.
The set is convex. Take any two points $a$ and $b$ in the unit square, consider the line joining them $a+t(b-a)$ where $t\in[0,1]$, and check that the components of every point on the line lie in $[0,1]$.
Okay so after consultation with @mzp, I discovered that my question was not clear enough. I was considering dictionary order topology, which I didn't make clear in the question or discussion. In that case, I $\times$ I is not convex.

Thank you so much everyone. I really appreciate your inputs.