Convergence spaces, pseudotopological spaces, etc.: what's the big picture? when do we encounter them and what are they good for?

I will attempt to at least partially answer my own question by saying something of what, after reading several papers, I understand about how pre- and pseudotopological spaces shed light on quotients of topological spaces. I still very much invite other answers or additions, clarifications or corrections to this one, which I am making community wiki. — Gro-Tsen

Main references for what follow will be:

• [BHL] := Bentley, Herrlich & Lowen, “Improving Constructions in Topology” (p. 3–20 in: Herrlich & Porst (eds.) Category Theory at Work (Bremen 1990), Heldermann (1991))

• [HLS] := Herrlich, Lowen-Colebunders & Schwarz, “Improving $\mathbf{Top}$: $\mathbf{PrTop}$ and $\mathbf{PsTop}$” (p. 21–34 in: Herrlich & Porst (eds.), op. cit.)

It is known (though perhaps not as much as it should be, and perhaps traumatizing to learn) that, while quotients of topological spaces exist, they do not behave as nicely as they could. Specifically:

• If $X \xrightarrow{f} Y$ is a quotient map of topological spaces, it may not remain so under restriction: that is, if $Y_0 \subseteq Y$ a subspace, and if $X_0 := f^{-1}(Y_0)$ with the induced topology, then $X_0 \to Y_0$ need not be a quotient map (see example 1 below). When this does hold (for a given $f$, for all $Y_0 \subseteq Y$) we say that $f$ is hereditarily quotient.

• Even if $X \xrightarrow{f} Y$ is a hereditarily quotient map of topological spaces, it may not remain a quotient map under pullback: that is, if $Y' \xrightarrow{g} Y$ is a continuous map and $X' := X \times_Y Y'$ is the set of $(x,y) \in X \times Y'$ such that $f(x) = g(y)$ (with the induced topology), it need not be the case that $X' \to Y'$ (taking $(x,y)$ to $y$) is a quotient map. (In fact, this need not hold even when $Y' = Y\times Z$ so that $X' = X\times Z$ and $X' \to Y'$ is just $f \times 1_Z$, see example 2 below.) When this holds (for a given $f$ for all $g$) we say that $f$ is universally quotient (or sometimes bi-quotient, see below).

Topological conditions can be given for a quotient map: $f$ is hereditarily quotient iff it is continuous and pseudo-open, where “pseudo-open” means that for every $y\in Y$ and open set $V$ in $X$ such that $V \supseteq f^{-1}(y)$, the direct image $f(V)$ is a neighborhood of $y$. And $f$ is universally quotient iff it is bi-quotient, meaning that it is continuous and that for every $y\in Y$ and open covering $(V_i)_{i\in I}$ of $f^{-1}(y)$ in $X$, there are finitely many $i_1,\ldots,i_m \in I$ such that $f(V_{i_1}) \cup \cdots \cup f(V_{i_m})$ is a neighborhood of $y$.

(For a proof of the equivalences, see [BHL], propositions 28 and 35. A proof of the second equivalence had already appeared in: Day & Kelly, “On topological quotient maps preserved by pullbacks or products”, Proc. Camb. Phil. Soc. 67 (1970) 553–558. The statement of the first equivalence had appeared without proof in: Arkhangel'skii, “Mappings and Spaces”, Russian Math. Surveys, 21 (1966), 115–162.)

The reason pretopological spaces and pseudotopological spaces shed light on the situation is the following:

• Regarding pretopological spaces:

  • Quotients of pretopological spaces exist (defined by putting the “obvious” pretopology in the set quotient) and they are hereditary. ([BHL], prop. 26; or [HLS], theorems 1.4 and 2.4.)

  • A continuous map of topological spaces, when considered as a continuous map of pretopological spaces, is a quotient map in the latter category iff it is hereditarily quotient in the category of topological spaces, i.e., by what has already been said, iff it is pseudo-open. ([BHL], prop. 28. This had already appeared as theorem 4(a) in: Kent, “Convergence Quotient Maps”, Fund. Math. *65 (1969), 197–205.)

  • If $X \to Y$ is a quotient map of pretopological spaces in which $X$ is a topological space, then the corresponding quotient of topological spaces (i.e. the topological quotient by the equivalence relation on $X$ whose set quotient is the underlying set of $Y$) is the topological reflection of $Y$. (The topological reflection of any flavor of convergence space is the one whose open sets are the $U$ such that if $\mathcal{F} \blacktriangleright x$ and $x \in U$ then $U \in \mathcal{F}$.) This holds for abstract nonsense reasons.

• Regarding pseudotopological spaces, the situation is nearly identical, mutatis mutandis, with quotients being even better behaved:

  • Quotients of pseudotopological spaces exist (defined by putting the “obvious” pseudotopology in the set quotient) and they are universal (i.e., preserved by pullback). ([BHL], prop. 33.)

  • A continuous map of topological spaces, when considered as a continuous map of pseudotopological spaces, is a quotient map in the latter category iff it is universally quotient in the category of topological spaces, i.e., by what has already been said, iff it is bi-quotient. ([BHL], prop. 35. This had already appeared as theorem 5 in: Kent, op. cit..)

  • If $X \to Y$ is a quotient map of pseudotopological spaces in which $X$ is a topological space, then the corresponding quotient of topological spaces is the topological reflection of $Y$; and the corresponding quotient of pretopological spaces is the pretopological reflection of $Y$. (The pretopological reflection of any flavor of convergence space is the one whose vicinity filter at $x$ is the intersection of all $\mathcal{F}$ such that $\mathcal{F} \blacktriangleright x$.) Again, this holds for abstract nonsense reasons.

To summarize: pretopological spaces, and pseudotopological spaces even more so, have nicely behaved quotients, and the quotients of topological spaces are obtained by performing the quotient in one of these categories and then relecting back to topological spaces (which spoils the niceness).

(There is something analogous for convergence1 spaces with “almost open” maps, see Kent, op. cit., definition 8 and theorem 4(b), but I'm not sure if all the exactly analogous statements hold.)

This means, in particular, that examples of badly behaved quotients of topological spaces provide interesting examples of pretopological or pseudotopological spaces (taking the corresponding quotient in the respective category) that are not topological spaces, whose topological reflection is the topological quotient. So I now present two examples of such situations.

Example 1: A quotient of topological spaces that is not hereditary, and that provides an example of a pretopological space that is not topological: consider the disjoint union $X = X_0 \amalg X_1$ of the half-open interval $X_0 = \mathopen]0,1\mathclose]$ and the set $X_1 = \{0\} \cup \{\frac{1}{n} : n\geq 1\}$ each one with its topology as a subspace of $\mathbb{R}$, and consider the quotient of $X$ by the equivalence relation that identifies $\frac{1}{n}$ in $X_0$ and $X_1$: if $\tilde Y$ is the quotient as a pretopological space and $Y$ as a topological space, both of these are structures on the set quotient $[0,1]$.

Around any point other than $0$, the pretopological vicinity structure of $\tilde Y$ and the topological neighborhood structure coincide and also coincide with the neighborhood structure from the usual topology on $[0,1]$. At $0$, however, they differ: the vicinity filter at $0$ in the pretopological quotient $\tilde Y$ is generated by the $\{0\} \cup \{\frac{1}{n} : n\geq m\}$, whereas the neighborhood filter at $0$ in the topological quotient $Y$ consists of subsets of $[0,1]$ that contain $0$ and some usual neighborhood of the $\frac{1}{n}$ for $n\geq m$ (for some $m\geq 1$).

(We can directly check that $X \to Y$ is not hereditarily quotient by restricting it to the subset $Y' = [0,1] \setminus \{\frac{1}{n} : n\geq 1\}$ of $Y$: then $0$ is isolated in $X'$ but not in $Y'$, so $Y'$ is not a quotient of $X'$.)

I think in this example the pseudotopological quotient is the same as the pretopological one, but I didn't check too carefully.

Example 2: A quotient of a topological space that is hereditary but not preserved under product, and that provides an example of a pseudotopological space that is not pretopological: let $X = \mathbb{R}$ with the usual topology, and consider its quotient by the equivalence relation that identifies every element of $\mathbb{Z}$ (call $z$ the resulting element).

If $\tilde Y$ is the quotient as a pseudotopological space and $Y$ as a pretopological space, then around every point other than $z$ both are just the usual topology on $\mathbb{R}$. At $z$, however we have $\mathcal{F} \blacktriangleright z$ in the pseudotopological quotient $\tilde Y$ when $\mathcal{F}$ is the image of a filter on $X$ that converges (in the usual sense) to some integer, whereas the vicinity filter $\mathcal{V}_z$ in the pretopological quotient $Y$ is the intersection of all such $\mathcal{F}$, namely the quotient image of the filter of open sets containing $\mathbb{Z}$ in $X$, i.e., it is generated by the $\bigcup_{k\in\mathbb{Z}} \mathopen]k-\varepsilon_k,k+\varepsilon_k\mathclose[$ (where all $\varepsilon_k$ are positive). This is also the neighborhood filter for the topological quotient, so here topological and pretopological quotients coincide.

We can check directly that $X\to Y$ is not preserved under product: $X \times \mathbb{Q} \to Y \times \mathbb{Q}$ is not a quotient map (see [BHL], example 4; or argue as follows: in $Y \times \mathbb{Q}$, a neighborhood of $(z,0)$ must contain a $\big(\bigcup_{k\in\mathbb{Z}} \mathopen]k-\varepsilon_k,k+\varepsilon_k\mathclose[\big) \times \mathopen]-\delta,\delta\mathclose[$ where $\varepsilon_k>0$ and $\delta>0$, whereas in the quotient of $X \times \mathbb{Q}$ by the equivalence relation that identifies all elements of each $\mathbb{Z}\times\{r\}$ must merely contain a $\bigcup_{k\in\mathbb{Z}} \big(\mathopen]k-\varepsilon_k,k+\varepsilon_k\mathclose[ \times \mathopen]-\delta_k,\delta_k\mathclose[\big)$).

To conclude, let me remark that every pseudotopological space is the quotient (in the category of pseudotopological spaces) of a topological space. (I didn't find this stated in the literature, so the proof below is mine, as are possible errors it may contain.)

Indeed, if $X$ is a pseudotopological space and $X_{\operatorname{disc}}$, resp. $X_{\operatorname{coarse}}$ its underlying set considered as a discrete resp. coarse topological space, and $\beta X_{\operatorname{disc}}$ the space of ultrafilters on $X$ (i.e., the Stone-Čech compactification of the discrete space $X_{\operatorname{disc}}$), we can consider the graph $\Gamma := \{(x,\mathcal{U}) : \mathcal{U} \blacktriangleright x\} \subseteq X_{\operatorname{coarse}} \times \beta X_{\operatorname{disc}}$ of the relation $\blacktriangleleft$ (inverse of $\blacktriangleright$). In fact, a pseudotopological space structure $X_{\operatorname{disc}} \to X$ is exactly defined by this subset $\Gamma$ subject to the sole constraint that it contains the graph of the “principal ultrafilter” map $X_{\operatorname{disc}} \to \beta X_{\operatorname{disc}}$.

Now $\Gamma$ can be considered a topological space with the induced topology (from the product of the coarse topology on $X_{\operatorname{coarse}}$ and the Stone-Čech compactification topology $\beta X_{\operatorname{disc}}$ of the discrete one), and one can check that an ultrafilter $\mathfrak{V}$ on $\Gamma$ converges to $(x,\mathcal{U}) \in \Gamma$ iff the second projection of $\mathfrak{V}$ converges to $\mathcal{U}$ (and of course $(x,\mathcal{U}) \in \Gamma$ imposes the constraint $\mathcal{U} \blacktriangleright x$).

So if we consider the pseudotopological space quotient of the topological space $\Gamma$ by projection to the first coordinate, we have $\mathcal{U} \blacktriangleright x$ in this quotient iff $(x,\mathcal{U}) \in \Gamma$ i.e. iff $\mathcal{U} \blacktriangleright x$ in the pseudotopological structure $X$ we started with, and $X$ is a pseudotopological quotient of the topological space $\Gamma$.

This means that the situation considered above is typical: pre- or pseudotopological spaces are always described by quotients of topological spaces.

Here are some observations that may be of use.

• Firstly I will add that a convergence2 space is sometimes said to be of finite depth. For me convergence0 is the basic concept of a convergence space and the others (Kent, of finite depth, pseudotopological, pretopological and topological) are properties that a convergence space may have. There are some comments about there being a maze of definitions, but to me it seems that these definitions are well-established and natural.

• To me the most interesting examples of non-topological convergences come from ordered spaces. In a complete lattice there is a very natural notion of convergence: a filter $F$ converges to $x$ if $$ \bigvee_{A\in F} \bigwedge A = x = \bigvee A \bigwedge_{A\in F}. $$ This is, of course, the usual $\liminf$-$\limsup$ notion of convergence. It is usually not topological. In fact it is in general only a Kent space, not even of finite depth. (For me this is a very compelling reason to take convergence0 as the basic definition). This definition can be extended to all posets by considering the embedding into the Dedekind-MacNeille completion and letting the poset have the initial convergence w.r.t. this embedding. This procedure turns out to have a very nice intrinsic definition as well: a filter $F$ on a poset $P$ converges to $x$ in this order convergence iff there exist non-empty sets $M,N \subseteq P$ such that $[a,b] \in F$ for all $a\in M, b\in N$ and $\bigvee M = x = \bigwedge N$. There are several other intrinsic ways of defining this convergence. See https://link.springer.com/article/10.1007/s11117-022-00885-2, for example.

• As an extension of the previous point, in Riesz spaces (i.e. vector lattices: vector spaces with a compatible order structure that is a lattice) it is common to define a notion of "order convergence". The usual definition is specific to Riesz spaces and usually phrased in terms of nets, but it turns out that it is equivalent to this order convergence. There are many somewhat ad hoc definitions in Riesz space theory that were given vaguely topological / convergence theoretical names, such as "order dense", "order closed", "order continuous", "order bounded" etc. Many of these notions turn out to be closely linked to the actual convergence theoretic notion associated with the order convergence. For example a Riesz ideal is "order dense" in ad hoc terminology iff it is dense w.r.t. the order convergence (i.e. the adherence it the whole space). Unfortunately the two notions diverge on more general subsets.

• There are several other settings where people have defined an ad hoc notion of convergence that can be connected to the more general theory of convergence spaces. The most famous is of course the notion of "convergence almost everywhere". This is not topological. Another is the notion of "algebraic interior" of a subset of a vector space. This is usually defined without reference to a convergence, but it turns out that it is actually the inherence connected to a fairly natural notion of convergence.

• It has been highlighted in the previous answer, but it bears repeating that in some ways the structural properties of convergence spaces are much nicer than those of topological space. Quotients are much nicer. For example the convergence quotient of a locally compact space is locally compact. This is not true for topological quotients. The category is also "Cartesian closed": if $X, Y$ are convergence spaces, then there exists a weakest convergence on $\mathcal{C}(X,Y)$ such that the evaluation map $\mathcal{C}(X,Y) \times X \to Y$ is continuous. This convergence is topological if $X$ is locally compact and $Y$ is topological and regular. In this case it is given by the compact-open topology. Really "Cartesian closed" just means that the properties of the compact-open topology can be generalised.

• Following on from the previous point (and indeed the previous answer) I will remark that in fact every Kent convergence space is the quotient of a topological space: let $X$ be a convergence space. For any $F \blacktriangleright x$ in $X$, define the convergence $\xi_{F,x}$ on $X$ by $\mathcal{N}(x) = F\cap \{x\}^\uparrow$ and $\mathcal{N}(y) = \{y\}^\uparrow$ (where $y\neq x$). This convergence is topological. Take the coproduct of all these convergences. This is still topological. Now take the quotient that identifies all the distinct copies of the set. We recover the original convergence.

• It is surprising to me how many theorems of topology still have some form that holds in the convergence space setting. The biggest difficultly seems to me the definition of separation properties beyong regularity. While regularity has a generally agreed upon definition in the convergence setting, anything stronger is a mess. Theorems also start breaking down. For example a non-topological compact Hausdorff space is no longer regular.

• In general the main obstruction to generalisation is that general convergence spaces may not have "enough" open and closed sets. To give just one example, a subset $A\subseteq X$ is called locally closed if, for every $x\in A$ there exists a vicinity $V_x$ of $x$ in $X$ such that $V_x \cap A$ is closed in $V_x$. In a topological space this is equivalent to $A$ being the difference of two open sets. In a general convergence space, these open sets may not exist (but if they do, then the set is locally closed). The concept of being locally closed is still relevant, however. For example, it is still true that a locally closed subset of a locally compact convergence space is locally compact. If $X$ is pretopological and Hausdorff, it is also true that if $A$ is locally compact, $A$ is also locally closed.

• One defect of non-topological spaces that may not be obvious is in the category of convergence vector spaces. It is no longer true that the vicinities filter at the origin has a balanced base. This impacts some results, in particular related to von Neumann boundedness. In a similar vein, if $V$ is a vicinity of the identity in a convergence group, we can no longer find a vicinity $W$ such that $W\cdot W \subseteq V$. For results in this setting I recommend the textbook by Beattie and Butzmann.

• Many famous theorems, such as Hahn-Banach separation and Krein-Milman also hold in a convergence setting.

Hopefully these points are of some use to somebody. I realise I have not been at all precise, but the answer is already long! The results should mostly be taken as intuition-building, but, of course, do ask if you want more details.