You may have a look at What is a continuous path? for a very much related discussion.

My opinion, in two words, is that the main property of $[0,1]$ is that one can glue the intervals and obtain basically the same thing. I think that one can define the fundamental group as soon as one has a way to describe paths using an ordered set for which the concatenation of intervals is well-behaved. Let me give an explicit example: in A-homotopy theory of graphs, one uses the natural numbers as ordered set and defines the continuous paths to be finitely supported functions from $\mathbb N$ to the graph with the property that $f(i-1)$ is a neighbor of $f(i)$. Then defines homotopy equivalence and gets a group that has several good properties. If you are interested, you may find details in Chapter 2 of http://arxiv.org/abs/1111.0268

I never really went through the details but I am confident that one can do something similar also using the hyperreal numbers as ordered set and define for instance the fundamental group of *$\mathbb R^2$ and *$\mathbb R^3$. I think that there is some hope to prove that they are not homeomorphic. This fact does not seem easy to prove using classical notions, as you can see here Non standard Algebraic Topology

You may have a look at What is a continuous path? for a very much related discussion.

My opinion, in two words, is that the main property of $[0,1]$ is that one can glue the intervals and obtain basically the same thing. I think that one can define the fundamental group as soon as one has a way to describe paths using an ordered set for which the concatenation of intervals is well-behaved. Let me give an explicit example: in A-homotopy theory of graphs, one uses the natural numbers as ordered set and defines the continuous paths to be finitely supported functions from $\mathbb N$ to the graph with the property that $f(i-1)$ is a neighbor of $f(i)$. Then defines homotopy equivalence and gets a group that has several good properties. If you are interested, you may find details in Chapter 2 of http://arxiv.org/abs/1111.0268

I never really went through the details but I am confident that one can do something similar also using the hyperreal numbers as ordered set and define for instance the fundamental group of *$\mathbb R^2$ and *$\mathbb R^3$. I think that there is some hope to prove that they are not homeomorphic. This fact does not seem easy to prove using classical notions, as you can see here Non standard Algebraic Topology

You may have a look at What is a continuous path? for a very much related discussion.

My opinion, in two words, is that the main property of $[0,1]$ is that one can glue the intervals and obtain basically the same thing. I think that one can define the fundamental group as soon as one has a way to describe the paths using an ordered set for which the concatenation of intervals is well-behaved. Let me give an explicit example: in A-homotopy theory of graphs, one uses the natural numbers as ordered set and defines the continuous paths to be finitely supported functions from $\mathbb N$ to the graph with the property that $f(i-1)$ is a neighbor of $f(i)$. Then defines homotopy equivalence and gets a group that has several good properties. If you are interested, you may find details in Chapter 2 of http://arxiv.org/abs/1111.0268

I never really went through the details but I am confident that one can do something similar also using the hyperreal numbers as ordered set and define for instance the fundamental group of *$\mathbb R^2$ and *$\mathbb R^3$. I think that there is some hope to prove that they are not homeomorphic. This fact does not easy to prove using classical notions, as you can see here Non standard Algebraic Topology

You may have a look at What is a continuous path? for a very much related discussion.

My opinion, in two words, is that the main property of $[0,1]$ is that one can glue the intervals and obtain basically the same thing. I think that one can define the fundamental group as soon as one has a way to describe paths using an ordered set for which the concatenation of intervals is well-behaved. Let me give an explicit example: in A-homotopy theory of graphs, one uses the natural numbers as ordered set and defines the continuous paths to be finitely supported functions from $\mathbb N$ to the graph with the property that $f(i-1)$ is a neighbor of $f(i)$. Then defines homotopy equivalence and gets a group that has several good properties. If you are interested, you may find details in Chapter 2 of http://arxiv.org/abs/1111.0268

I never really went through the details but I am confident that one can do something similar also using the hyperreal numbers as ordered set and define for instance the fundamental group of *$\mathbb R^2$ and *$\mathbb R^3$. I think that there is some hope to prove that they are not homeomorphic. This fact does not seem easy to prove using classical notions, as you can see here Non standard Algebraic Topology