Let $X$ be a set and let $\Phi(X)$ denote the collection of [filters](https://en.wikipedia.org/wiki/Filter_(mathematics)) on $X$. For $x\in X$ we denote by $P\_x$ the filter $P\_x=\{A\subseteq X:x\in A\}$. A *convergence space* is a pair $(X,\to)$, where $X$ is a set, and $\to$ is a subset of $\Phi(X)\times X$ with the following properties:


1. $P\_x \to x$ for all $x\in X$ (we write ${\cal F}\to x$ instead of $({\cal F},x)\in \to$), and
2. If ${\cal F}\subseteq {\cal G}\in \Phi(X)$ and ${\cal F}\to x$, then ${\cal G}\to x$.


If $(X,\to\_X)$ and $(Y,\to\_Y)$ are convergence spaces, then $f: X\to Y$ is said to be *continuous* if ${\cal F}\in \Phi(X), x\in X$ and ${\cal F}\to\_X x$ imply $f({\cal F}) \to\_Y f(x)$, where $f({\cal F})$ is the [image filter](https://en.wikipedia.org/wiki/Filter_(mathematics)#Functions_on_topological_spaces) of ${\cal F}$.


The category ${\bf Conv}$ consists of convergence spaces with continous maps between them. There is a functor ${\bf Top}\to {\bf Conv}$ constructed in the following way:


* Objects: $(X,\tau)$ maps to $(X,\to\_\tau)$ where ${\cal F} \to\_\tau x$ if and only if ${\cal F} \supseteq {\cal N}\_x$, where ${\cal N}\_x$ is the neighborhood filter of $x$ in $(X,\tau)$.
* Maps: It is easy to see that a continous map in the topological sense is continuous in the convergence space sense, so the functor is the identity on the morphisms.


**Question.** Does this functor have an adjoint going back from ${\bf Conv}$ to ${\bf Top}$?




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**Note.** Many topological properties can be transferred to convergence spaces. For instance, a convergence space $(X,\to)$ is *compact* if for every ultrafilter ${\cal U}\in \Phi(X)$ there is $x\in X$ such that ${\cal U}\to x$. Also, [Tychonoff's theorem](https://en.wikipedia.org/wiki/Tychonoff%27s_theorem) can be proved for compact convergence spaces. Moreover, a convergence space is said to be $T\_2$ if ${\cal F}\to x$ and ${\cal F} \to y$ imply $x=y$.



Adjoints for the functor ${\bf Top}\to {\bf Conv}$
At @DominicvanderZypen's [request](https://mathoverflow.net/questions/283376/adjoints-for-the-functor-bf-top-to-bf-conv#comment699664_283376), here is
@მამუკა-ჯიბლაძე's answer from the comments [1](https://mathoverflow.net/questions/283376/adjoints-for-the-functor-bf-top-to-bf-conv#comment699473_283376) [2](https://mathoverflow.net/questions/283376/adjoints-for-the-functor-bf-top-to-bf-conv#comment699478_283376) (marked as CW to avoid reputation):



> 
> In a comment to a [related question](https://mathoverflow.net/q/265596) Todd Trimble gave a link to the nLab entry on [relational $\beta$-modules](https://ncatlab.org/nlab/show/relational+beta-module) which I think answers much of this question.
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> It [the construction of the functor $\mathbf{Conv} \to \mathbf{Top}$] is at that nLab page, just [before Proposition 2.1](https://ncatlab.org/nlab/show/relational+beta-module#bridge_to_a_concrete_description): $U$ is open iff for any $\mathcal F \to x$ with $\mathcal F$ an ultrafilter, $x \in U$ implies $U \in \mathcal F$.
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