Connected Metric Space Question

In summary, the question at hand is whether for a connected metric space X, with distinct points a and b, and a given radius r, there exists a finite collection of open balls of radius r that is connected and contains both a and b. The proposition of being r-ball-connected is introduced and the idea of considering the set of points that are r-ball connected to a is suggested. The question of whether this set is both open and closed is raised.
  • #1
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Let X be a connected metric space, let a, b be distinct points of X and let r > 0. Is there a collection {B_i} of finitely many open balls of radius r such that their union is connected and contains a and b.

I was trying to prove this by contradiction, but couldn't derive a contradiction. I can't think of any counterexamples either. Does anyone know whether this statements is true or false?
 
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  • #2
Hrm. Well, let's say that a and b are r-ball-connected if such a collection exists.

A common argument regarding variants of connectedness is to consider something like

A(a,r) = the set of all points that are r-ball connected to a.

Now, A(a,r) is clearly an open set. Is it closed? (I'm assuming you know the ramifications of a set being both open and closed)
 
  • #3
Great tip. Thanks for clearing that up.
 

1. What is a connected metric space?

A connected metric space is a mathematical concept that describes a set of points that are connected in a specific way. In a connected metric space, any two points can be connected by a path that remains within the space and does not break or disconnect at any point.

2. How is connectedness defined in a metric space?

Connectedness in a metric space is defined by the property that all points in the space can be connected by a path. This means that if any point is chosen as a starting point, and any other point is chosen as an end point, there exists a continuous path that connects the two points while remaining within the space.

3. What is the importance of connected metric spaces in mathematics?

Connected metric spaces are important in mathematics because they provide a way to understand the structure of a set of points and how they are connected to each other. They also play a crucial role in many areas of mathematics, including topology, analysis, and geometry.

4. Can a metric space be connected and disconnected at the same time?

No, a metric space cannot be both connected and disconnected at the same time. A metric space is either connected or disconnected, based on the definition of connectedness. If any two points can be connected by a path, the space is connected. If there exists a separation between points, the space is disconnected.

5. How are connected metric spaces used in real-world applications?

Connected metric spaces have various real-world applications, such as in physics, biology, and computer science. In physics, connectedness is used to describe the continuity of space and time. In biology, connectedness is used to understand the relationships between different organisms and their environments. In computer science, connected metric spaces are used in data analysis and network theory.

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