In a directed graph, certain paths are distinguished: Those that follow the edge direction on each edge. I was wondering if one could (or does) formulate a notion of a 'Directed Topological Space', on which certain paths are distinguished. Conceptually, we could take a space like R^2 \ 0 and say that counterclockwise paths are the 'directed paths'.

I had two ideas: First, if you have a smooth Riemannian manifold, then a smooth vector field X gives you a 'Direction': The directed paths being those whose tangent vector v satisfies g(v,X) > 0 everywhere.

Alternatively, with an arbitrary topological space, you could say that a 'Directed' space comes equipped with a sorting of homotopy classes of paths into 'Forward', 'Backward', and 'Trivial' (the last consisting of just trivial loops), where the compositions 'Forward'*'Forward' = 'Forward', 'Forward'*'Trivial' = 'Forward', etc. and the reverse of a 'Forward' path is always a 'Backward' path (note that you wouldn't have any results involving 'Foward'*'Backward' or 'Backward'*'Forward').

So I suppose my question is: Is this idea studied at all? Has anyone determined this (or perhaps another generalization of 'Directed') to be a useful notion?


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    $\begingroup$ ncatlab.org/nlab/show/directed+topological+space $\endgroup$ – Aaron Mazel-Gee Mar 13 '13 at 21:27
  • $\begingroup$ Perhaps this is not what you are looking for, but your question reminds me of the notion of the exit-path category of a stratified space. This is similar to the notion of the fundamental groupoid of a space $X$: the objects are points of $X$ and the morphisms are paths. However, in the exit path category, we only allow paths that move from lower dimensional strata to higher. Consequently, we get a category rather than a groupoid. $\endgroup$ – Sam Gunningham Mar 14 '13 at 20:19

There is quite a growing number of researchers nowadays in so-called "Directed Algebraic Topology" (it even has its own Wikipedia page). It can be seen as part of Applied Algebraic Topology, since it arose out of an attempt to model concurrent systems in Computer Science.

I can recommend the papers of Fajstrup, Raussen and Goubault (linked to from the Wiki page) and Jeremy Gunawardena on homotopy and concurrency as good introductions to the subject of directed homotopy.

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Partially ordered sets are closely related to directed graphs, and partially ordered spaces have been studied in general topology quite a bit since there is a strong connection between ordered sets and topologies. I realize that this may not be exactly what you had in mind since you may have been concerned mainly with paths on such spaces, but I will answer this question since ordered topological spaces are very interesting and very natural. Besides, one may also consider paths on ordered topological spaces as order preserving continuous maps from $[0,1]$ to your space $X$. Furthermore, I suppose that you can probably generalize this notion to a "locally ordered topological space" or a "locally preordered topological space" so that you can consider $\mathbb{R}^{2}\setminus\{0\}$ with a notion of going clockwise and counterclockwise. I will briefly outline some basic ordered topological spaces here.

If $X$ is a topological space, then the specialization ordering on $X$ is the preordering $\leq$ on $X$ such that $x\leq y$ if and only if $x\in\overline{\{y\}}$. Furthermore, the specialization ordering on $X$ is a partial ordering if and only if $X$ is a $T_{0}$-space. It follows from the definition that $X$ is a $T_{1}$-space if and only if the specialization ordering $\leq$ coincides with equality. Thus the specialization ordering is next to useless for spaces satisfying separation axioms from $T_{1}$ and above. However, the specialization ordering is very important for analyzing spaces that do not satisfy the $T_{1}$-separation axiom.

As another example of an ordered topological space, an Alexandroff space is a topological space where the arbitrary intersection of open sets is open. The category of Alexandroff spaces is isomorphic to the category of all preordered sets where the morphisms between preordered sets are the order preserving maps. If $X$ is an Alexandroff space, then the specialization ordering gives you a preordering on $X$. If $X$ is a preordered set, then the collection of all downwards closed sets are precisely the closed sets in an Alexandroff topology on $X$. These correspondences give you the equivalence between the category of Alexandroff spaces and the category of partially ordered sets.

Also, people have endowed partially ordered sets with other topologies such as the Lawson topology and Scott topology (see the book Continuous Lattices and Domains or the book the Compendium of Continuous Lattices) for more details. The Scott topology almost never satisfies higher separation axiom, but it turns out that the Lawson topology is compact and Hausdorff on a large class of ordered sets called continuous lattices.

Other ordered spaces that satisfy higher separation axioms are also useful. For instance, one can use ordered compact Hausdorff spaces in order to study distributive lattices. We say that a topological space $X$ with a partial order $\leq$ is totally order disconnected if whenever $x\not\leq y$, then there is an upwards closed clopen set $U$ with $x\in U,y\not\in U$. A Priestley space is a compact totally order disconnected topological space. It turns out that the category of Priestley spaces is equivalent tot the category of all ordered topological spaces.

In my own personal research I have studied connections between least upper bounds in posets and topological spaces, and this research is quite similar to the Scott topology on a topological space. In my research, I have shown that a certain category of zero-dimensional ordered topological spaces is isomorphic to a category certain of pairs of the form $(X,\mathcal{A})$ where $\mathcal{A}$ is a collection of subsets of $X$ with least upper bounds.

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  • $\begingroup$ How are posets "closely related to directed sets"? $\endgroup$ – Vidit Nanda Mar 14 '13 at 0:48
  • $\begingroup$ I changed "directed set" to "directed graph." $\endgroup$ – Joseph Van Name Mar 14 '13 at 2:18
  • $\begingroup$ In the same vein, ordered simplicial complexes might be of interest to the OP. mathoverflow.net/questions/100610/… $\endgroup$ – Lee Mosher Mar 14 '13 at 15:15

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