Connectedness in the plane There are several open problems in topology which concern connectedness and subsets of the plane.  The biggest of these is undoubtedly:

Question. Does every non-separating plane continuum have the fixed point property?
This is a very beautiful problem.  In essence, it is asking whether
  every non-separating plane continuum is like a closed disk, or
  conversely, if every separating plane continuum is like a circle.

Many other open problems are not explicitly about subsets of the plane, but it helps to think about whether they are true for spaces that can be embedded in the plane.  The plane is, in many cases, the nicest embedding space that does not make a problem trivial (the real line lacks sufficient complexity). 
In this post I would like to compile a list of topological properties of the plane.  They should be fairly easy to state, and it is perfectly okay to state a result in (non-trivially) different ways. 
I know that this is a general request, but it would be extremely helpful to me. 

Jordan Curve Theorem. Every simple closed curve separates the plane into two components.
Denjoy–Riesz Theorem. Every compact totally disconnected subset of the plane is contained in an arc.  
(Kuratowski). If a plane continuum $X$ is the common boundary of three
  of its complementary components, then $X$ is either indecomposable or the
  union of two proper indecomposable subcontinua.
(Rudin). If there are three arcs in the plane which do not contain the point $x$ and which have only their end-points in common, then one of these arcs is separated from $x$ by the union of the other two. 

Nice theorems of the geometric variety are also welcome.

Sylvester–Gallai Theorem. Given a finite number of points in the
  plane, either all the points are collinear, or there is a line which
  contains exactly two of the points.
Erdős–Anning Theorem An infinite number of points in the
  plane can have mutual integer distances only if all the points lie on
  a straight line.

 A: My favourites:


*

*Janiszewski’s theorem: Let $A$ and $B$ be two compact subsets of the complex plane and $a, b \not\in A \bigcup B$. If neither $A$ nor $B$ separates $a$ and $b$ and if $A \bigcap B$ is connected, then $A \bigcup B$ does not separate $a$ and $b$.

*A bounded planar domain is simply connected if and only if its complement is connected if and only if its boundary is connected.

*(Schoenflies): A closed planar set $F$ is a Jordan curve if and only if its complement has two components, and each point of $F$ is accessible from both of the components.

*(R.L.Moore): A bounded domain is a Jordan domain if and only if it is simply connected and uniformly locally connected.
I also suggest to add good references on the plane topology. I know Whyburn -  Topological Analysis, Newman - Elements of the Topology of Plane Sets of Points. Apparently R.L. Moore - Foundations of Point Set Theory contains a lot of material, but I haven't really read it due to rather nonstandard writing. Also Pommerenke - Univalent Functions and also Boundary Behaviour of Conformal Maps have nice sections on the plane topology.
Might add something later...
A: The Jordan curve theorem was strengthened by Schoenflies to the statement that the two components are homeomorphic to the inside and outside of a circle.  The Alexander horned sphere shows that this strengthening is not true in three dimensions.
In graph theory, Kuratowski showed that a graph is planar if and only if it contains no subgraph that is a subdivision of $K_5$ or $K_{3,3}$.
A: Here is a pretty recent result by John Pardon on the continuous deformation of simple rectifiable closed curves to convex ones, without decreasing the distance between any two points. The result was already proved for polygonal closed curves.
A: The Annulus Theorem: If $J_1$ is a Jordan curve in the interior region of another Jordan curve $J_2$, then the complement of the exterior region of $J_2$ and the interior region of $J_1$ is the closed annulus.
The Torhorst Theorem: If $X$ is a Peano continuum in the plane and $B$ is a complementary region, then $\partial(B)$ is a Peano continuum.
The Moore Triod Theorem: If $X$ is a triodic continuum then there is no uncountable collection of mutually disjoint copies of $X$ in the plane.
The Roberts Embedding Theorem: If $X$ is an arc-like or circle-like continuum in the plane, then there is an uncountable collection of mutually disjoint copies of $X$ in the plane.
Classification of Homogeneous Planar Continua: A point, a circle, a pseudo-arc, and a circle of pseudo-arcs.
The Isotopy Extension Theorem: Any isotopy between compact $A$ and $B$ extends to an isotopy on the plane.
Tarski Circle-Squaring Theorem: A closed disc can be decomposed into finitely many pieces, then reassembled into a closed square by translating the pieces.
As for open problems, how about the Mandelbrot Conjecture, the Siegel Disc Conjecture, the Toeplitz Square problem . . . for some 'smaller' ones how about a characterization of planar dendroids, of continua which admit uncountably many copies in the plane, or the classification of isotopy classes of arc-like continua based on their inverse limit representations?
A: Perhaps this just-posted note, which settles an earlier conjecture:

Y.G. Nikonorov, Y.V. Nikonorova,
"One property of a planar curve whose convex hull covers a given convex figure."
arXiv abstract.
1 July 2020.


If the convex hull of a planar curve $\gamma$ covers a planar convex figure $K$, then $$\operatorname{length}(\gamma)\ge \operatorname{per}(K) − \operatorname{diam}(K)\;.$$


          


