Let $X$ be a topological space, we say it is $T_D$ if each of its points is locally closed. That is, for any $x\in X$ we have $\lbrace x\rbrace=U\cap V$ where $U\subseteq X$ is an open subset and $V\subseteq X$ is closed.
This is a separation axiom lying between $T_0$ (the Kolmogorov axiom) and $T_1$. Indeed, if $X$ is $T_D$ and we have two distinct points $x,y$, it is easy to provide an open subset containing $x$ but not $y$. Conversely, if the space is $T_1$, by definition each singleton is closed, hence locally closed.
My question is the following: what are the implications of the $T_D$ axiom, especially regarding locally closed subsets in general? That is, can we deduce that if a subspace of $X$ is locally closed, then it has some properties? Or, vice versa, that a subspace of $X$ having a certain property must be locally closed?
For example, it is know that the finite intersection of locally closed subsets is still locally closed. While the union (even finite) of locally closed subsets is not necessarily locally closed. If the space is $T_D$, can we say something on the line of "if we have an arbitrary family of locally closed subsets, all of which have property P, then their intersection is locally closed" or "if two subspaces have property P and are locally closed, then their union is again locally closed"?
I know the question is very vague, but from what I understood this $T_D$ property, other than being a curious separation axiom between $T_0$ and $T_1$, does not have many relevant applications. Since I am not an expert in point set topology I suspect this is false and this property has some usefulness that I am not aware of.
Best Answer
In the context of spectral spaces, the $T_D$ axiom does occur naturally.
A spectral space is a topological space which is
That is quite a mouthful, but spectral spaces are precisely those topological spaces which arise as the Zariski spectra of commutative rings. This is a theorem of Hochster proved in
Another context in which they arise is in pointless topology. One version of Stone duality is that the category of spectral spaces (and spectral maps) is anti-equivalent to the category of coherent frames (and coherent frame maps) which in turn is equivalent to the category of bounded distributive lattices. Under these equivalences, a spectral space $X$ corresponds to the bounded distributive lattice of its quasi-compact open sets (ordered by inclusion). Indeed, a subset of the above axioms says that the quasi-compact open subsets form a bounded distributive lattice. One can read more about this in
although he uses the term "coherent space" for "spectral space".
In any case, a wealth of information about spectral spaces can be found in the book:
A Thomason subset of a spectral space $X$ is a subset $Y\subseteq X$ which is a union of closed sets, each of which has quasi-compact complement. The Thomason subsets form the open sets for a "dual" spectral topology on $X$ (called the inverse topology in [Dickmann et al.]). The resulting spectral space $X^*$ is called the Hochster dual of $X$. (Under Stone duality, taking the Hochster dual amounts to taking the opposite order on the corresponding bounded distributive lattice.)
In the study of support theories of big tensor-triangulated categories $\mathcal T$, one is led to the following topological hypothesis on a certain spectral space $X=\mathrm{Spc}(\mathcal T^c)$ called the Balmer spectrum of compact objects of $\mathcal T$. Namely, we say that a spectral space $X$ is weakly noetherian if each point $x \in X$ can be expressed as $\{ x\}=Y_1 \cap Y_2^c$, where $Y_1,Y_2 \subseteq X$ are Thomason subsets. (Every noetherian spectral space is weakly noetherian.) The reason this hypothesis turns up is because to each Thomason subset $Y \subseteq \mathrm{Spc}(\mathcal T^c)$ of the Balmer spectrum there is an associated pair of (co)localization functors on $\mathcal T$; morally the colocalization restricts $\mathcal T$ to the part of the category supported on $Y$, while the localization restricts $\mathcal T$ to the part of the category supported on the complement $Y^c$. Thus, the weakly noetherian hypothesis morally ensures that you can "isolate" the part of the category lying over the point $x$ as a composite of a localization and a colocalization, and this leads to a definition of "support" for the objects of $\mathcal T$. See
Now, a spectral space is "weakly noetherian" in the above sense precisely when its Hochster dual is $T_D$. As someone working in this area of mathematics, the $T_D$ condition is thus actually very natural and useful in my work. See also
for an explicit mention of the $T_D$ condition in the literature (related to what I have sketched above).
Further places where this topological condition arises can be found in the references in 4.5.10 of [Dickmann et al.]