Prove that $\{(x, y)\in \mathbb{R^2}:x<y\}$ is connected.

Question

Prove that $\{(x, y)\in \mathbb{R^2}:x<y\}$ is connected.

Looking at the graph it is clear but how do I proof it theoretically. Thank you

Answer 1

If $(x,y)$ belongs to your set and $t\in[0,1]$, then every point of the form $\bigl((1-t)x,t+(1-t)y\bigr)$ also does, because$$t+(1-t)y-(1-t)x=t+(1-t)(y-x)>0.$$Therefore, your set is path connected, because the map\begin{array}{rccc}\gamma\colon&[0,1]&\longrightarrow&\left\{(x,y)\in\mathbb{R}^2\,\middle|\,y>x\right\}\\&t&\mapsto&\bigl((1-t)x,t+(1-t)y\bigr)\end{array}is a path from $(0,1)$ to $(x,y)$. Since it is path-connected, it is connected.

Answer 2

Let $A$ be the rotation to $\pi/4$ clockwise, then $(x,y)\rightarrow A(x,y)$ is homeomorphism from $S$ to the upper half plane $M:=\{(x,y): y>0\}$.

It suffices to prove that $M$ is path-connected. But this is clear because any point in the line segment of any two points of $M$ must have strictly positive $y$-coordinate. Think of the convexity.