Trait Suggestion: Topologist's sine curves S113, S114, S115 are not Cut point spaces P205, but they have a cut point P204

[Moniker1998]

Trait Suggestion

The spaces S113, S114, S115 are not Cut point spaces P205, but they have a cut point P204, but this fact is not known to pi-Base today

Proof/References

If $X$ is any of the spaces S113, S114, S115 then $(0, 0)$ is not a cut point of $X$, since $X\setminus \{(0, 0)\}$ is connected.

But its not a cut point space since taking any $x\in (0, 1)$ in case of S113 and S114 and $x\in (0, 1)$ close enough to $1$ in case of S115, we see that $X\setminus\{(x, \sin(1/x))\}$ is not connected.

[GeoffreySangston]

@Moniker1998 I think part of your argument shows $(0, 0)$ is non-cut point, and every other point is a cut point P204 I believe. So asserting Has a cut point and ~Cut point space P205 will imply ~homogeneous.

[Moniker1998]

Yes, that definitely does work.

S113, S114 and S115 all have property P204 but not P205