Fundamental groups and arcwise connected spaces.

[center o bass]

If a space X is arcwise connected, then for any two points p and q in X the fundamental groups $\pi_1(X,p)$ and $\pi_1(X,q)$ are isomorphic. This means that we can, up to isomorphisms, identify both groups with their equivalence class $\pi_1(X)$.

I started to think about the generality of this theorem. Is there any interesting/important spaces which are not arcwise connected except for the union of two disjoint spaces etc?

 

[Mandelbroth]

If a space X is arcwise connected, then for any two points p and q in X the fundamental groups $\pi_1(X,p)$ and $\pi_1(X,q)$ are isomorphic. This means that we can, up to isomorphisms, identify both groups with their equivalence class $\pi_1(X)$.

I started to think about the generality of this theorem. Is there any interesting/important spaces which are not arcwise connected except for the union of two disjoint spaces etc?

Any Hausdorff space that is not path-connected is also not arc-connected. Therefore, the so-called "Topologist's Sine Curve," a relatively important topological space, is not arc-connected.

 

[micromass]

Any Hausdorff space that is not path-connected is also not arc-connected. Therefore, the so-called "Topologist's Sine Curve," a relatively important topological space, is not arc-connected.

It's not a relatively important topological space, since it serves more as a counterexample than as something worth studying on its own. Sure, it's a nice and important counterexample, but that's it. It's not a topological space central to algebraic topology like topological manifolds.

 

[Mandelbroth]

It's not a relatively important topological space, since it serves more as a counterexample than as something worth studying on its own. Sure, it's a nice and important counterexample, but that's it. It's not a topological space central to algebraic topology like topological manifolds.

Meh. It was the first Hausdorff space that isn't path-connected that came to mind.

I have to agree with micro, though. It isn't important itself. Come to think of it, off the top of my head, I can't think of any particularly "important" spaces that are not arc-connected.

As a side note, topological manifolds are locally arc-connected (since they are Hausdorff and locally path-connected).

 

[mathwonk]

what is the definition of "important"? what about the rational numbers? some people think they are important. we need more details to answer this, i think.