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 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 of 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