Proving the Connectedness Math Problem in R^n: Is it True?

  • Thread starter kingwinner
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In summary, the statement "If S is connected, then the interior of S is connected" is true. Although the "backward" case of starting with an open set and considering its closure does not necessarily yield the same result, it is still a subset of the possibilities and can help with finding a proof or counterexample. Taking the example of an interval (a,b] or [a,b), it is clear that the interior of [a,b] and (a,b) are connected, but the interior of [a,b) is not necessarily connected. This generalization holds for R^n as well. Therefore, the statement is true but it is important to consider all possibilities and not just specific examples.
  • #1
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"Let S be a subset of R^n.
If S is connected, then the interior of S is connected."

Is this true or not?

I can't think of a counterexample, but I don't know how to prove it either...
 
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  • #2
kingwinner said:
"Let S be a subset of R^n.
If S is connected, then the interior of S is connected."

Is this true or not?

I can't think of a counterexample, but I don't know how to prove it either...

It might help to think "backwards" -- rather than thinking of starting with a set S, and then working with its interior, why not start with an open set, and then consider its closure (possibly excluding part of the boundary)?
 
  • #3
But that "backward" one is not equivalent to the original one.

How can I find a counterexample?
 
  • #4
True, the backward case is just a subset of the possibilities -- but I assert that it covers enough of the possibilities that it should suggest a proof or yield a counterexample.
 
  • #5
I think you can take something like
an interval (a,b] or [a,b) (the same should work also for R^n).
obviously if [a,b) is connected, then (a,b) is'nt necessarily.
 
  • #6
Is the claim true or false?
I can't figure it out and I don't know which direction to push my proof towards...
 
  • #7
It seems true with simple examples, but we can't generalize from specifics. How can I start the proof in the general situation?
 
  • #8
loop quantum gravity said:
I think you can take something like
an interval (a,b] or [a,b) (the same should work also for R^n).
obviously if [a,b) is connected, then (a,b) is'nt necessarily.
?? All of [a,b], [a,b), (a,b], (a,b) are connected!
 

1. What is a "Connectedness math problem"?

A Connectedness math problem is a type of mathematical puzzle where the goal is to find the minimum number of connections needed to connect a set of points or objects together. This type of problem is commonly used in graph theory and computer science.

2. How do you solve a Connectedness math problem?

The solution to a Connectedness math problem involves identifying the minimum number of connections needed to connect all the points or objects together. This can be done by drawing a diagram and counting the number of connections needed, or by using mathematical formulas and algorithms to find the minimum number of connections.

3. What are some real-life applications of Connectedness math problems?

Connectedness math problems have many practical applications in fields such as transportation, computer networking, and social networks. For example, in transportation planning, these problems can help determine the most efficient routes for buses or trains to connect different locations.

4. Are there different types of Connectedness math problems?

Yes, there are various types of Connectedness math problems, including the minimum spanning tree problem, the shortest path problem, and the traveling salesman problem. Each type has its own specific rules and methods for solving.

5. Is there a specific strategy for solving Connectedness math problems?

There is no one specific strategy for solving Connectedness math problems, as each problem may require a different approach. However, some common strategies include breaking the problem into smaller, manageable parts, using visual aids such as diagrams or graphs, and using mathematical formulas and algorithms.

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