Euler characteristic of $\mathbb{R}$ and $\mathbb{R}^2$?

I´m trying to understand the value of the Euler characteristic of the real line and the real plane.

I don´t know if it is defined, I think that it is for any topological space.

So this could be right?

If we separate $\mathbb{R} = (-\infty,x_0] \cup [x_0,x_1] \cup [x_1,+\infty)$

$\mathcal{X}(\mathbb{R}) = V - E + F = 2 - 3 + 0 = -1$

Analogously, considering a "triangle" in the plane,

$\mathcal{X}(\mathbb{R}^2) = V - E + F = 3 - 3 + 2 = 2$

Thanks!

NOTE: I´ve seen the related topic:

Is the Euler characteristic of $\mathbb{R}^n$ $1$ or $(-1)^n$?

But I didn´t understand it.

Any reasonable definition of the Euler characteristic should satisfy the product property $$ \chi(N\times M)=\chi(N)\chi(M). $$ This is violated here in your calculation for $N=M=\Bbb R$.