connected region with piecewise smooth curves

Question

why $D=\mathbb{R}^2$ $-\{(x,0):x\in\mathbb{R}\}$ is not connected but $D=\mathbb{R}^3$ $-\{(x,0,0):x\in\mathbb{R}\}$ is connected?( according to the definition of connected regions by piecewise smooth curves)

Answer 1

If you need to use the formal definitions, it may be useful to include the ones you're using.

For some intuition:

why $D=\mathbb{R}^2$ $-\{(x,0):x\in\mathbb{R}\}$ is not connected

Imagine the infinite plane and color the $x$-axis red: you cannot go from a point above the $x$-axis to a point below this axis without crossing the red line; right?

but $D=\mathbb{R}^3$ $-\{(x,0,0):x\in\mathbb{R}\}$ is connected?

Now imagine the infinite (three dimensional) space and again, color the $x$-axis (only the axis; not a plane!) red. Can you think of two points that aren't reachable without crossing the red line?