Fundamental groups and arcwise connected spaces.

In summary, the theorem states that 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. However, there are Hausdorff spaces that are not path-connected, such as the "Topologist's Sine Curve," which is not arc-connected. This space is not considered important in itself, but it serves as a useful counterexample. It is not a topological space central to algebraic topology like topological manifolds. The definition of "important" in this context is not specified. However, some may argue that spaces like the rational numbers are important.
  • #1
center o bass
560
2
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?
 
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  • #2
center o bass said:
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.
 
  • #3
Mandelbroth said:
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.
 
  • #4
micromass said:
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. :rolleyes:

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).
 
  • #5
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.
 

Related to Fundamental groups and arcwise connected spaces.

1. What is the fundamental group of a space?

The fundamental group of a space is a mathematical concept that measures the number of holes or "handles" in a space. It is denoted by π1(X), where X is the space, and is a group of all possible loops in the space starting and ending at a fixed point.

2. How is the fundamental group of a space calculated?

The fundamental group of a space can be calculated using algebraic topology techniques such as the Van Kampen theorem or the Seifert-van Kampen theorem. These theorems allow us to break down a space into simpler pieces and then combine their fundamental groups to obtain the fundamental group of the original space.

3. What is an arcwise connected space?

An arcwise connected space is a topological space in which any two points can be connected by a continuous path or arc. This means that there are no "holes" or "gaps" in the space, and it is possible to move from one point to another without leaving the space.

4. How are arcwise connected spaces related to the fundamental group?

Arcwise connected spaces are important in the study of the fundamental group because they ensure that the fundamental group is well-defined and does not depend on the choice of basepoint. In other words, for arcwise connected spaces, the fundamental group is the same at every point in the space.

5. What are some real-world applications of fundamental groups and arcwise connected spaces?

Fundamental groups and arcwise connected spaces have many applications in areas such as physics, biology, and computer science. For example, in physics, the fundamental group can be used to understand the topology of spacetime, and in biology, it can be used to study the shape of protein molecules. In computer science, fundamental groups can be used for data compression and in the development of algorithms for path planning and optimization.

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