Property Suggestion
A space is said to be $T_D$ (or $T_{\frac{1}{2}}$) if for every $x \in X$, there is an open neighborhood $U \ni x$ such that $U\setminus\{x\}$ is also open. There are also a bunch of other equivalent formulations:
- for every $x$ there is a neighborhood s.t. $U \cap \overline{\{ x \}}=\{ x \}$
- for every $x$ the derived set $\{x\}'$ is closed
- probably more
Rationale
This property appears to be assumed quite a lot in pointfree topology and is closely related to several notions of sobriety. It's initial definition is by Aull and Thron in "Separation Axioms Between T0 and T1" (1962). It has since been mentioned in "Frames and Locales" by Jorge Picado and Aleš Pultr in section 2.1, and on nLab.
The rationale for adding it is that it was brought up several time on MSE like in 5097648, 3179291 - example of a topological space that is $T_{D}$ but not sober, and 5148449 - where the asker specifies that they haven't found the property on pi-base. That sounds like a feature request to me :)
Relationship to other properties
- $T_1 \implies T_D$
- $T_D\implies T_0$
- Alexandrov + $T_0 \implies T_D$ (as per comment by Ulli)
- certainly some more, I will add them
More sources
https://topology.lmf.cnrs.fr/td-spaces/ - great exposition with equivalent formulations and implications by Jean Goubault-Larrecq.
"Non-Hausdorff separation axioms" by Tianyi Zhou, mentions $T_D$ space in section 7:
https://arxiv.org/pdf/2511.18527#section.7
https://ecommons.udayton.edu/cgi/viewcontent.cgi?article=1054&context=topology_conf - slides mentioned in the MSE post, also mentions the "hereditarily sober" property, might worth looking into.
Let me know if this is something pi-base will be interested in, or whether something equivalent already exists.
Property Suggestion
A space is said to be$T_D$ (or $T_{\frac{1}{2}}$ ) if for every $x \in X$ , there is an open neighborhood $U \ni x$ such that $U\setminus\{x\}$ is also open. There are also a bunch of other equivalent formulations:
Rationale
This property appears to be assumed quite a lot in pointfree topology and is closely related to several notions of sobriety. It's initial definition is by Aull and Thron in "Separation Axioms Between T0 and T1" (1962). It has since been mentioned in "Frames and Locales" by Jorge Picado and Aleš Pultr in section 2.1, and on nLab.
The rationale for adding it is that it was brought up several time on MSE like in 5097648, 3179291 - example of a topological space that is$T_{D}$ but not sober, and 5148449 - where the asker specifies that they haven't found the property on pi-base. That sounds like a feature request to me :)
Relationship to other properties
More sources
https://topology.lmf.cnrs.fr/td-spaces/ - great exposition with equivalent formulations and implications by Jean Goubault-Larrecq.
"Non-Hausdorff separation axioms" by Tianyi Zhou, mentions$T_D$ space in section 7:
https://arxiv.org/pdf/2511.18527#section.7
https://ecommons.udayton.edu/cgi/viewcontent.cgi?article=1054&context=topology_conf - slides mentioned in the MSE post, also mentions the "hereditarily sober" property, might worth looking into.
Let me know if this is something pi-base will be interested in, or whether something equivalent already exists.