Nonseparating planar continuum

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A nonempty compact set $K\subseteq\mathbb R^2$ is path connected if any two points are joined by a continuous map $[0,1]\to K$. It is nonseparating if $\mathbb R^2\setminus K$ is connected. It has the fixed-point property if every continuous map $f:K\to K$ fixes some point. Continuity uses the subspace topology.

Statement recorded in the supplied source.

Does every nonempty path-connected nonseparating compact set $K\subseteq\mathbb R^2$ have the fixed-point property? Explicitly, does \[ \forall f\in C(K,K)\quad\exists x\in K:\quad f(x)=x \] follow from those hypotheses? [S2]

Short English statement

Must a continuous self-map of a path-connected planar compact set with connected complement have a fixed point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3423 (OPG-37295). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Nonseparating planar continuum, OPG-37295. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/nonseparating_planar_continuum.