What is a continuous path?

Question

I would like some help, because I am getting mad trying to answer the following

Question: Let $X$ be a topological space, what is a continuous path in $X$?

Well, maybe you're already getting nervous thinking: it's just a continuous function $\gamma:[0,1]\rightarrow X$. This definition indeed works very well for manifolds and, more generally, for spaces containing homeomorphic copies of the interval $[0,1]$, but it gets trivial and useless in all other cases, including some of great interest: graphs and, more generally, locally finite metric spaces in the discrete world, but also non-standard objects as ${}^*\mathbb R$.

At some point, I've realized a very stupid thing; namely that the gap in the classical definition of a continuous path is that the notion of continuity is imposed from outside, taking as a unit of measurement the unit interval $[0,1]$. This is quite arbitrary, isn't it? This observation was somehow revolutionary, at least for me: at that point, I closed my eyes, imaging to live in a topological space and I tried to capture a notion of continuity from inside: a natural answer is that it would sound, roughly, like: continuity is to move from one point to another one doing the shortest possible steps...

This philosophical definition can be made formal for a quite general class of metric spaces (containing for instance all locally finite connected graphs)

Example: Let $(X,d)$ be a locally finite metric space. Given $x\in X$, denote by $dN_1(x)$ the smallest closed ball about $x$ which contains at least two points. One may define a continuous path in $X$ to be a sequence of points $x_0,x_1,\ldots,x_{n-1}x_n$ such that, for all $i$, $x_i\in dN_1(x_{i-1})$ and $x_{i-1}\in dN_1(x_i)$. Following a similar idea, one is tempted to define homotopy between paths and so on..

Everything works unexpectedly well as you can see, if interested, in http://arxiv.org/abs/1111.0268. As remarked by Tim Porter, similar ideas have been developed by Helene Barcelo and co-authors.

What I would like to do now is to approach the problem of defining an intrinsic homology theory that might be of interest for any topological space.

Subquestion: do you know if somebody tried to do something similar?

In case of a negative answer to the sub-question, I would appreciate also any help finding the answer to the first question. Indeed, I am really satisfied from the locally finite case and I would like to formalize the philosophical definition: a continuous path connecting $x$ to $y$ is a way to go from $x$ to $y$ making the shortest possible steps. But it is absolutely not clear to me how to make it formal for a general topological space.

Thanks in advance,

Valerio

Update: In case someone is interested, some of these ideas got finally accepted for publication in a paper with Jacob White and Helene Barcelo in the Bull London Math Soc. http://arxiv.org/pdf/1306.3915.pdf

Answer 1

It sounds to me like what you're looking for is something like Cech (co)homology. The idea is that you can detect what kind of "paths" there are in a space by the combinatorics of which sets in open covers have nontrivial intersections. As a simple example, you can detect that a circle has a nontrivial loop by covering it with 3 open sets $U$, $V$, and $W$, such that any two of them intersect but $U\cap V\cap W$ is empty. More precisely, given any open cover, you can construct a simplicial complex which is the "nerve" of the open cover and has the "paths" that a space having that open cover "morally" should have. Of course, you shouldn't expect a single open cover to capture all of the information you're trying to capture about a space, so you have to take some sort of limit over all open covers of your space. Taking finer and finer open covers is like taking the "shortest possible steps" that you refer to.

As an example of how this might give you what you're looking for, Cech cohomology can't tell the difference between an ordinary circle and a loop built out of a topologist's sine curve, or an ordinary circle and a circle obtained by gluing together the two ends of a closed long line. It won't work for things like the hyperreals unless you restrict to some sort of internal open covers, because the hyperreals as a topological space are disconnected, and Cech cohomology detects topological connectedness (but not path-connectedness in the usual sense). And of course, if you're dealing with things like locally finite metric spaces which are discrete as spaces, there's no hope of saying anything interesting unless you endow the spaces with more structure than just a topology.

As a technical point, Cech cohomology is pretty well-behaved, but if you try to do the same thing with homology you run into problems because when you take a limit over all open covers you end up taking an inverse limit of homology groups, and taking inverse limits is not exact. According to nLab there is something called strong homology which tries to remedy this which you might want to take a look at.

EDIT: If you want something that can work for discrete spaces with additional structure, note that you can take the nerve of one specific covering, rather than taking a limit over all covers. For instance, I would guess that your homology theory of locally finite metric spaces is the same as the Cech homology of the cover consisting of the balls $dN_1(x)$ for all $x$. For "nice" spaces a similar phenomenon occurs: the limit over all covers coincides with what you get from a single cover satisfying some simple condition. For example, for simplicial complexes, you can take the cover consisting of the open star of each vertex, or for a Riemannian manifold, you can take a cover consisting of geodesically convex sets.

Answer 2

Eric, Tim, and David have basically given what I think is the right answer, but I want to mention the words "site" and "topos" which have surprisingly not yet occurred in this discussion, and give some more references.

A site or topos is a generalization of a topological space, built on the notions of part and cover. (A topos is the true object; a site is a sort of "base" or "presentation" which generates a topos.) Roughly, a topos has a collection of parts, with containment relations between them, and a notion of when a part is covered by some collection of parts contained in it. This is sufficient for Cech-style homology and homotopy theories, and the resulting notion of "path" is indeed the sort of thing you have described: chains of parts which meet each other.

Every topological space gives rise to a topos whose parts are its open subsets, where covers are unions. This is the canonical way of embedding topological spaces into toposes. (It is a full embedding for most topological spaces.) But a topological space can also give rise to other toposes. For instance, one can consider the topos whose parts are the arbitrary (not necessarily) open subtoposes of the topos above; this topos is called the "dissolution" of a space. Or going in the other direction, one can take the open parts but allow only good covers.

I think this is the proper way to deal with the question of disconnectedness. One should not, I feel, expect to find any "paths" in a very disconnected object: by the very fact of its being disconnected, there should not be any ways to get from here to there. If one sees "paths" in a disconnected topological space, that suggests that there is some data other than the topology defining some other object in which the paths live. For instance, to take Eric's example, the hyperreals give rise to a topos whose parts are the internal open subsets, and this topos will (I believe) be connected.

A couple of interesting papers which are specifically about paths in this context are:

• Kennison, What is the fundamental group?
• Moerdijk and Wraith, Connected locally connected toposes are path-connected.

Like much other work in this area, they restrict to the locally connected case, but the ideas do not really require that restriction (things just get more complicated otherwise).

Answer 3

Adding into the comments by Alain Valette: ... Hence my suggestion of Strong Shape Theory. Vietoris homotopy gives one way of approaching strong shape. Again it is mentioned on the nLab. Vietoris homology still has the problem of non-exact sequences. It can be replaced by Steenrod-Sitnikov homology or Mardesic's strong homology theory. (There is also a renewal of interest in finite topological spaces, see work by Minian and Barmak. An application of similar ideas occurs through topological data analysis, see work by Carlsson et al at Stamford. This also uses a Rips complex, that is probably known to you from your general interests.)

Looking at some of the other comments and answers, it may help to look at some of the ideas of graph homotopy theory that are around. These are linked via the original work of Dowker (1953) on the homology of a relation. (I can provide more indicators if you think it would help.)

(Edit: The following describes a related theory: Perspectives on A-homotopy theory and its applications, Hélène Barcelo, Reinhard Laubenbacher, Discrete Mathematics 298 (2005) 39 – 61.)

Answer 4

The technique/framework mentioned by Jim Conant and used by Berestovskii and Plaut goes back to the paper

• J. Krasinkiewicz and P. Minc, Generalized paths and pointed 1-movability, Fundamenta Mathematicae 104 (1979), 141-153, doi:10.4064/fm-104-2-141-153.

For an update on this subject see Rips complexes and covers in the uniform category, and the last section in Steenrod homotopy (which includes a simplified proof of the Krasinkiewicz-Minc result).

Edit: I was writing this in rush for an airplane, and could not elaborate on what the references are about. Now that strong shape (and even related things like Cech cohomology and Steenrod-Sitnikov homology) have been mentioned by others this simplifies my job.

What Krasinkiewicz and Minc were doing in that paper is essentially paths in the sense of strong shape. (They don't explicitly speak of "strong shape", but on the other hand it happens that papers and books that originally developed strong shape, including Tim Porter's, and most of subsequent literature under the "strong shape" brand has been incredibly focused on either categorical or general-topology aspects and didn't care to pursue any specific geometric problems, so if you're interested in any kind of substantial results on paths in the sense of strong shape, you have to look for them elsewhere!)

In the above-mentioned paper, Krasinkiewicz and Minc proved the following wonderful theorem: If $X$ is a connected (metrizable) compactum that is disconnected in the sense of strong shape (that is, not all strong shape morphisms from a point into $X$ are the same) then there exist distinct strong shape morphisms (in fact, uncountably many ones) from a point into $X$ that are represented by genuine points in $X$. This may sound like it should be either trivial or wrong, but no, it's a deep geometric result.

Answer 5

Instead of going to the lake (actually the weather is not that good), I've spent all Sunday morning on your references. First of all, thank you very much everybody. Here I want to collect some comments about Cech cohomology and the approach by Berestovskii and Plaut. I have just seen that Tim Porter and Alain Valette have suggested to look at something else. My afternoon will be devoted to those references.

Cech cohomology: as observed even by Eric himself, Cech cohomology does not work for disconnected spaces, so my first thought was that the answer didn't help. Indeed I am now sure (well, OK, I'm too young to be sure about something!) that there is some homology/homotopy/cohomology/whatever-theory completely general and intrinsic that is of interest (=non trivial) for any topological space. This is why I cannot accept the answer, but I give +1 because I like the point of view and I think that, at the end, the bad behavior of Cech cohomology is given by the fact that it uses open coverings. The notion of open set is too over-used (does this English word exist??) and sometimes one needs something different (an example is the van Kampen theorem). For instance, in a locally finite metric space every set is open and so it is clear that they are too many. So I give +1 because I want to have a closer look to Cech cohomology in order to understand is one can replace the open coverings with something that works better.

Berestovskii and Plaut: At the beginning I got really scared, because I've thought they were doing the same things. But NO. I have to say that I disagree with their approach for the following reasons: besides the (important, but in this case, minor) problem that the construction depends on the radius of the entourage (so it is not clear what happens when the radius goes to zero in a locally finite metric space, or what should be the right radius to choose and so on), I thing that the problem is the definition of homotopy between $\epsilon$-chains. Recall their

Definition: An $\epsilon$-chain is a finite sequence $x_0,x_1,\ldots,x_{n-1}x_n$ such that $d(x_i,x_{i-1})<\varepsilon$, for all $i$. Two $\epsilon$-chains are called homotopic equivalent if one can pass from the first to the second via a finite sequence of operation of adding/cancelling point in such a way that every intermediate step is an $\epsilon$-chain starting from $x_0$ and ending in $x_n$.

One obtains a group and bla bla bla. The point is that this construction is not interesting for instance for finite graphs (one gets the usual free group generated by the missing edges of a spanning tree)

What I have proposed in my preprint is the following (Update: it is turned out that (very) similar definitions have been alreadyproposed in the so-called $A$-theory (see references in Tim Porter answer or in my OT).

Definition: Two continuous paths (in the sense of my OT) $x_0x_1\ldots x_{n-1}x_n$ and $y_0y_1\ldots y_{n-1}y_n$ (I can suppose that the length is equal adding some constant path), with $x_0=x_n=y_0=y_n$ are homotopic equivalent if one can find points $z_i^k$ such that the following formal matrix

$$ \left( \begin{array}{ccccc} x_0 & x_1 & \ldots & x_{n-1} & x_0 \\\ x_0 & z_1^2 & \ldots & z_{n-1}^2 & x_0 \\\ \ldots & \ldots & \ldots & \ldots & \ldots \\\ x_0 & z_1^{k-1} & \ldots & z_{n-1}^{k-1} & x_0 \\\ x_0 & y_1 & \ldots & y_{n-1} & x_0 \\\ \end{array} \right) $$

verifies the property that every row and every column is a continuous path.

This definition, which seems to me more natural (roughly, a continuous deformation of a path is to replace each point of the path to one of the nearest point), also works incredibly well and non trivially. For instance we (my two collaborators for the second piece, A. Gournay and T.Pillon, and I) have an example of a graph with 28 vertex whose fundamental group is $\mathbb Z_2$. I'd like to include it here (you may be interested), but I have no idea how to include a figure here.

Answer 6

There is something like Cech cohomology. It is Alexander-Kolmogorov cohomology (see Spenyer "Algebraic topology", Alexander cohomology). Simplixes in this theory are collections of near points. As I understand it works well for locally contractible spaces (and coincides with Cech cohomology). But it seems that in your situation it works well too.

Answer 7

It sounds to me like your inventions are related to persistent homology, developed by Weinberger, Carlsson, and others. There is an informative "What is..." article about this by Weinberger: http://www.ams.org/notices/201101/rtx110100036p.pdf. The idea is to take a discrete subset of Euclidean space and calculate its homology on all different scales essentially by covering it with balls of a given radius $R$ and analyzing what happens as $R$ varies. If your space has a cycle in it then this cycle will be detected for a certain range of values of $R$, and the assumption is that the important structures will "persist" for more values of $R$.

Another idea which might be related is Roe's coarse geometry. There's a "What is..." article about this as well: http://www.ams.org/notices/200606/whatis-roe.pdf. Here the interest is in infinite discrete spaces (or really any non-compact metric space), and there is a coarse cohomology theory which detects only the large-scale geometry of a space.

Answer 8

Another possible direction on the fundamental group(oid) is given a kind of survey in my paper

`Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions', Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland) 8.05.2005-15.05.2005, Banach Centre Publications 76, Institute of Mathematics Polish Academy of Sciences, Warsaw, (2007) 51-63. (math.DG/0602499).

relating to the themes of monodromy and holonomy, and work of Jean Pradines. Here one is interested in the notion of an "iteration of local procedures" in the context of manifold theory. The question put in that paper is: Can one use these ideas in other situations to obtain monodromy (i.e. analogues of "universal covers") in situations where paths do not exist but "iterations of local procedures" do? This seems related to Valerio's question.

Answer 9

I am completely layman here, but I have found that question because I thought about very the same idea as OP. At first I've asked myself, why such discrete object as combinatorial simplex can describe something continuous as toplogical torus for example. And my answer was as follows: because homology theory use simplicial sets in general position, it means that only intersections of various n dimensional simplexes matters. I mean we may reduce all the theory to equivalent, using 0 dimensional elements ( points), and various sets which represents more complicated simplexes. For example line segment is just a discrete set (1,2) where 1 and 2 are just ends. In this representation, path is just a ordered set family when subsequent sets have non empty intersections. For discrete spaces it may be even restrictions to the size of such intersections: it should be set of the strictly minimal cardinality. For 2element sets it should consist only 1 element sets, for 3 element sets, only 2 element sets etc. Complex in such setting is a set consisting several subsets, with various cardinality. For example ( combinatorial) triangle would be: (1,2,2,(1,2),(2,3),(3,1),(1,2,3)). In such setting, a hole in a complex is situation where there is"closed path" ( first and least sets are the same) but there is no set in which elements are sum of the sets consisting the path. I suppose it follows to something which is very similar to Čech homology but I have no knowledge to decide if it is.

What is interesting it should not be very complicated to use it to discrete objects like graphs. One should just consider not only points, but various sets of points, and for the set families building a paths, a sets is sums of its elements together ( as bigger cardinality objects, constructing something like simplexes of larger cardinality but inside discrete object). Please notice that for a given discrete space ( such as graph) one may consider various subgraphs immersed inside, and ask if they are as dense as possible or have holes ( sidestep of existing graph vertices, but not containing it). In such approach, we may think about internal geometry of subgraphs of given graph relating it to base graph as full space, and subgraphs as objects living inside, and having various characteristics. I would like to ask you for information if anything similar was done for discrete settings, and from combinatorial point of view?