Path-connected space

Definition

A path-connected space is a topological space in which any two points can be connected by a continuous path. This means that for any two points in the space, there exists a continuous function from the closed interval [0, 1] into the space that maps 0 to one point and 1 to the other. Understanding path-connectedness is essential for exploring concepts like connectedness and is also pivotal in applying results like Van Kampen's theorem, which deals with the fundamental group of spaces formed by joining path-connected components.

5 Must Know Facts For Your Next Test

1. Path-connectedness implies connectedness, meaning if a space is path-connected, it cannot be split into separate pieces.
2. A simple example of a path-connected space is the Euclidean space $$ extbf{R}^n$$, where any two points can be joined by a straight line.
3. Not all connected spaces are path-connected; there are examples like the 'topologist's sine curve' which illustrate this distinction.
4. In the context of Van Kampen's theorem, path-connected spaces play a crucial role as they ensure the existence of paths needed to combine fundamental groups.
5. Path-connectedness can be established using homotopies, which show how one continuous path can be transformed into another without leaving the space.

Review Questions

• How does the concept of path-connectedness relate to connectedness in topological spaces?
  • Path-connectedness directly implies connectedness since if you can connect any two points with a continuous path, you can't separate the space into disjoint open sets. However, the reverse isn't always true; some spaces may be connected but lack the ability to form continuous paths between every pair of points. Therefore, while all path-connected spaces are connected, not all connected spaces are necessarily path-connected.
• Describe how Van Kampen's theorem utilizes path-connected spaces in its application to fundamental groups.
  • Van Kampen's theorem states that if a space can be represented as a union of two path-connected open sets with a non-empty intersection, the fundamental group of the entire space can be computed from the fundamental groups of these sets. The requirement for path-connectedness ensures that loops can be constructed within each set and that paths exist to connect these loops through their intersection, allowing us to piece together the overall structure of the fundamental group.
• Evaluate an example where a topological space is connected but not path-connected, and discuss its implications for understanding fundamental groups.
  • An example of a space that is connected but not path-connected is the 'topologist's sine curve.' This space consists of points along the curve given by $$y = ext{sin}(1/x)$$ for $$x > 0$$ along with points along the line segment connecting $$(-1, 0)$$ to $$(0, 0)$. While it can't be split into disjoint open sets, there’s no way to draw a continuous path between points on the sine curve and those on the line segment without leaving the space. This highlights that while we can have one-piece spaces in terms of connectedness, their complexity might affect how we compute their fundamental groups using tools like Van Kampen’s theorem.

Related terms

Connected space: A connected space is a topological space that cannot be divided into two disjoint non-empty open sets, indicating that it is in one piece.

Continuous function: A continuous function between topological spaces is one where the pre-image of every open set is also open, ensuring there are no abrupt changes or breaks.

Fundamental group: The fundamental group is an algebraic structure that captures the number of distinct loops based at a point in a topological space, providing insight into its shape and connectedness.