Understanding Disconnectedness in Countable Metric Spaces

In summary, the conversation discusses the concept of total disconnection in discrete metric spaces. It is stated that every discrete metric space with at least 2 points is totally disconnected, but there is confusion about the statement and its equivalent forms. The conversation concludes that the truth of the statement is independent of what happens for 2 points in a countable metric space.
  • #1
Bachelier
376
0
We know that every discrete metric space with at least 2 points is totally disconnected.
Yet I read this:
A MS that is countable with more than 2 pts is disconnected. Is it that I'm misreading this statement. It sounds like if it has 2 or less points it is connected?
more means greater than.
 
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  • #2
Well, a metric space with 2 points is disconnected. So the statement should be: a countable metric space with 2 or more points is disconnected.
 
  • #3
what is the equivalent statement of:

no discrete MS with more than 2 pts is connectd?
 
  • #4
Every discrete space with more than 2 points is disconnected?
 
  • #5
micromass said:
Every discrete space with more than 2 points is disconnected?

Exactly, which is false. because of we have a discrete MS X with 2 pts it is also disconnected. Yet the statement asks every discrete MS with > 2 pts.
 
  • #6
That doesn't make the statement false! If I say: for all numbers n greater than 2 it holds that n+0=n, then this statement is true. Whether it holds for other numbers n does not change the validity of the statement.

So the truth of the statement "every countable metric space with more than 2 points is disconnected" is independent of what happens for 2 points!
 
  • #7
micromass said:
That doesn't make the statement false! If I say: for all numbers n greater than 2 it holds that n+0=n, then this statement is true. Whether it holds for other numbers n does not change the validity of the statement.

So the truth of the statement "every countable metric space with more than 2 points is disconnected" is independent of what happens for 2 points!

Makes sense. thanks
 

FAQ: Understanding Disconnectedness in Countable Metric Spaces

What is a discrete metric space?

A discrete metric space is a mathematical structure in which the distance between any two points is either 0 or 1. In other words, the metric space only contains isolated points, and there is no continuity in the space.

What is the difference between a discrete metric space and a continuous metric space?

The main difference between a discrete metric space and a continuous metric space is in the way the distance between points is defined. In a discrete metric space, the distance between any two points is either 0 or 1, while in a continuous metric space, the distance between points can take on any real value. This means that a discrete metric space does not have any concept of continuity, while a continuous metric space does.

How is a discrete metric space useful in mathematics?

Discrete metric spaces are useful in many areas of mathematics, including topology, analysis, and geometry. They allow for the study of discrete structures and provide a foundation for understanding more complex spaces.

What are some examples of discrete metric spaces?

Examples of discrete metric spaces include the set of natural numbers with the discrete metric, the set of integers with the discrete metric, and the set of all finite subsets of a given set with the discrete metric.

What are the applications of discrete metric spaces in science?

Discrete metric spaces have applications in various fields of science, such as computer science, biology, and physics. They are used in computer networks, DNA sequence analysis, and modeling of particle interactions, among others.

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