pi-base
Trait Suggestion: Closed Topologist's Sine Curve S114 has Fixed point property P89
Trait Suggestion
The space Closed Topologist's Sine Curve S114 has Fixed point property P89, but those facts are not known to pi-Base today: link to pi-Base
Proof/References
Let $X$ be space S114 and $C = \{(x, \sin(1/x)) : x\in (0, 1]\}\subseteq X$ and $D = X\setminus C$. Since image of a path-connected set is path-connected, $f(C)$ and $f(D)$ need to be contained in $C$ or $D$.
If $f(D)$ is contained in $D$, then since $D\cong [0, 1]$ has fixed point property, $f$ has a fixed point in $D$. If $f(C)$ is contained in $D$, then since $C$ is dense in $X$ and $D$ is closed, $f(D)$ is contained in $D$, so again $f$ has a fixed point in $D$. Lastly, if both $f(C)$ and $f(D)$ are contained in $C$, then $f(X)$ is contained in $C$, and $X$ being a connected compact set, $f(X) = I\subseteq C \cong (0, 1]$ is homeomorphic to a compact interval or a singleton, and so has the fixed point property. It follows that $f$ has a fixed point in $I$. In any case, $f$ has a fixed point, and so $X$ has the fixed point property.
This is already known, in Infinite-dimensional topology of function spaces by van Mill it exists as exercise A.12.10. I don't really know a good reference, but its probably there somewhere. Either way, the proof is simple.
@felixpernegger thanks
Speaking of, https://topology.pi-base.org/spaces/S000115/properties/P000089 It's unknown if extended topologist sine curve has this property
Speaking of, https://topology.pi-base.org/spaces/S000115/properties/P000089 It's unknown if extended topologist sine curve has this property
Yeah this is in my PR