R^ 2 with dictionary order topology is metrizable
101986
Prob. 2, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: The dictionary order topology on $\mathbb{R} \times \mathbb{R}$ is metrizable.
16517
Arcwise connected part of $\mathbb R^2$
12060
Homeomorphism to graph
100437
Classification of connected subsets of the real line (up to homeomorphism)
5448
Finding the closure of some subsets of the ordered square
127491
Isometric isomorphism between $R^2$ and $R$
152478
Is there a homeomorphism between the $n$-dimensional cube to the $n$-dimensional simplex that doesn't create hard corners?
114523
The coarsest such topology on $\mathbb{R^{2}}$ is the usual topology
2575
Showing that $\mathbb{R}$ is connected
147793
A variation on the induced topology (subspace topology)
35626
Is $(\mathbb{R}^2, lexicographic) \cong (\mathbb{R}, discrete)\times (\mathbb{R}, usual)$?
32420
A complete orthonormal system contained in a dense sub-space.
157578
perfect map in topology
23837
Why is $\mathbb{R}/{\sim}$ not first countable at $[0]$, where $x \sim y \Leftrightarrow x = y\text{ or }x,y \in \mathbb{Z}$?
150796
Subspace topology and order topology (2)
3217
Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers
122110
Character of the subobject classifier in the topos of $\mathbf{M}$-sets.