Are simply connected subsets of $\mathbb{R}^2$ all contractible?
60746
Simply connected does not imply contractible. Is there a nice counter example in $R^2$?
22464
Which of the following subsets of   $\mathbb{R}^2$ are compact
120623
Unit sphere in $\mathbb{R}^\infty$ is contractible?
116866
$A\times B$ connected component implies $A,B$ connect components
131890
Totally disconnected space that is not $T_2$
10541
How to prove Mandelbrot set is simply connected?
127826
Is the geometric realization of a pointed category contractible?
84189
Is $\mathbb{C}$ equal to $\mathbb{R}^2$?
14345
If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected?
31266
Proof of "the continuous image of a connected set is connected"
154009
Closed, simply connected manifolds which are not spheres
18572
Union of a family of connected pairwise not disjoint subspaces is connected
152451
REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
39076
Contraction of compact sets
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
100437
Classification of connected subsets of the real line (up to homeomorphism)
90826
Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly