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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.

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  • $\begingroup$ Is it a topological property that causes the rain in Spain to stay mainly in the plane? $\endgroup$ – Gerry Myerson May 8 '17 at 5:51
  • $\begingroup$ This is perhaps too simple, but it does distinguish the plane from Euclidean spaces of other dimensions: If you delete any point, the resulting space is connected but not simply connected. $\endgroup$ – Timothy Chow May 8 '17 at 22:36
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    $\begingroup$ I do not get the point that you seem to be trying to make: "or conversely, if every separating plane continuum is like a circle." The Warsow circle (also called $\sin\frac1x$ circle) is a separating plane continuum with the FPP (unlike the usual circle that doesn't have the FPP). books.google.com/… $\endgroup$ – Mirko May 23 '17 at 1:15
  • $\begingroup$ @Mirko correct. What I wrote hopefully gives a taste of what the problem is about, but is definitely an over-simplification. $\endgroup$ – Forever Mozart May 23 '17 at 5:47
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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...

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  • $\begingroup$ Those are very nice! I've come across the first one before, but I can't remember where. $\endgroup$ – Forever Mozart May 8 '17 at 1:55
  • $\begingroup$ You can find the proof in Pommerenke, or else in Newman under the name "Alexander Lemma". $\endgroup$ – erz May 8 '17 at 2:09
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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 $K_{3,3}$.

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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.

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