Topologically speaking, how is the set $\{(x, y) \in \mathbb{R}^2| y = 0\}$ related to $\mathbb{R}$ Topologically speaking, how is the set $A = \{(x, y) \in \mathbb{R}^2| y= 0\}$ related to $B = \mathbb{R}$
I am curious if there is some formal way to relate the topological properties of  $\{(x, y) \in \mathbb{R}^2| y= 0\}$ related to $\mathbb{R}$, say via some continuous bijective map. Also, how to map open sets from $A$ to $B$

$A$, the $x$-axis, is a topological copy inside $\Bbb R^2$ (even an isometric one) of $\Bbb R$ in the usual topology. The obvious homeomorphism to see this is $f: \Bbb R \to A; f(x)=(x,0)$ which is even an isometry ($d((x,0),(x',0)) = |x-x'|$ when $d$ is the Euclidean distance on $\Bbb R^2$ (but also when $d$ is the supremum distance or the taxi-cab metric). So not only topological properties of $A$ and $\Bbb R$ are the same ((path-)connectedness, local compactness, local connectedness etc.) but also metric properties (completeness, e.g.).
