Basic topology proof of closed interval in R

In summary, the problem statement is asking to prove that the union of a set of closed intervals in R, denoted by \bigcup\limits_{j \in J} {\{ [{a_j},{b_j}]\} }, is also a closed interval in R. However, the proof requires additional information about the type of subsets used in the disjoint union for this statement to hold true.
  • #1
lurifax1
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Let [tex] \{ [a_j, b_j]\}_{j\in J} [/tex] be a set of (possibly infinitely many closed intervals in R whose intersection cannot be expressed as a disjoint union of subsets of R. Prove that [tex] \bigcup\limits_{j \in J} {\{ [{a_j},{b_j}]\} }[/tex] is a closed interval in R.

I don't understand how to attack this, and would appreciate an example or a push in the right direction! This is the first proof exercise I'd had in this course and I'm pretty lost.
 
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  • #2
I think your problem statement is incomplete. Every subset of R can be expressed as a disjoint union of sets. I think the problem needs to say disjoint union of some particular kind of sets. What kind?
 

FAQ: Basic topology proof of closed interval in R

What is topology and how does it relate to the closed interval in R?

Topology is a branch of mathematics that studies the properties of geometric spaces that are preserved under continuous transformations. The closed interval in R, often denoted as [a,b], is a fundamental concept in topology as it represents a closed and bounded set in the real number line.

2. What is the significance of proving the closed interval in R in basic topology?

Proving the closed interval in R is important in basic topology as it serves as a foundation for understanding more complex topological concepts and theorems. It also allows for the development of techniques and skills that are essential in topology and other branches of mathematics.

3. What are the key steps in a basic topology proof of the closed interval in R?

The key steps in proving the closed interval in R in basic topology include defining the interval, showing that it is closed and bounded, and proving that it satisfies the three axioms of a topological space: the set is non-empty, the set itself is open, and the set is closed under finite intersections and arbitrary unions.

4. Can the closed interval in R be proven using other topological concepts?

Yes, the closed interval in R can also be proven using other topological concepts such as compactness, connectedness, and completeness. These concepts provide alternative approaches to proving the closed interval in R and help deepen our understanding of the topological properties of the interval.

5. How is the proof of the closed interval in R relevant in real-world applications?

The closed interval in R has many real-world applications, particularly in physics, engineering, and economics. For example, it is used in the study of motion and velocity in physics, in analyzing the stability of structures in engineering, and in optimizing production and pricing strategies in economics.

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