Subdivisions/Refinement Proof

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In summary: Therefore, D1 u D2 is a refinement of D1. In summary, if each of D1 and D2 is a subdivision of [a,b], then D1 u D2 is a subdivision of [a,b] and is also a refinement of D1.
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TheyCallMeMini
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Homework Statement



If each of D1 and D2 is a subdivision of [a,b], then...
1. D1 u D2 is a subdivision of [a,b], and
2. D1 u D2 is a refinement of D1.

Homework Equations



**Definition 1: The statement that D is a subdivision of the interval [a,b] means...
1. D is a finite subset of [a,b], and
2. each of a and b belongs to D.


**Definition 2: The statement that K is a refinement of the subdivision D means...
1. K is a subdivision of [a,b], and
2. D is a subset of K.

The Attempt at a Solution



I just proved that, "If K is a refinement of H and H is a refinement of the subdivision D of [a,b], then K is a refinement of D." Well I haven't wrote it down but the 2nd definition part 2 is what makes it easy to relate just being a transitive proof.

My problem is that I've taken a lot of logic courses in the past so when I see the union of two variables I only need to prove that one is actually true. In this particular situation both are true so its obvious but I don't know how to state that fact.

For the 2nd part of the proof, wouldn't I just say that D1 is a subset of itself, and its already given that D1 is a subdivision of [a,b]? It just seems too easy...I also had questions about proofs I've already turned in that I did poorly on but I didn't want to flood this place with questions.
 
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TheyCallMeMini said:
For the 2nd part of the proof, wouldn't I just say that D1 is a subset of itself, and its already given that D1 is a subdivision of [a,b]? It just seems too easy...

Yes, I agree. Since by the first part you will have already shown that D1 u D2 is a subdivision of [a,b], all you have to do in part 2 is simply state that D1 is a subset of D1 u D2, thus the definition of refinement is satisfied.
 

1. What is a subdivisions/refinement proof?

A subdivisions/refinement proof is a mathematical proof technique used in topology to show that a given space can be divided into smaller, simpler parts while still preserving its overall structure and properties.

2. How is a subdivisions/refinement proof used?

A subdivisions/refinement proof is used to break down a complex topological space into simpler components, making it easier to analyze and understand. It is also used to prove that two spaces are topologically equivalent.

3. What are the key steps in a subdivisions/refinement proof?

The key steps in a subdivisions/refinement proof involve defining a partition of the given space, showing that the partition satisfies certain conditions (such as being a cover or a refinement), and then proving that the partition preserves the topological properties of the original space.

4. What are some common challenges in a subdivisions/refinement proof?

One common challenge in a subdivisions/refinement proof is finding an appropriate partition that satisfies the necessary conditions. Another challenge is showing that the partition preserves the topological properties of the original space, which may require careful analysis and construction of the partition.

5. Can subdivisions/refinement proofs be used in other areas of mathematics?

Yes, subdivisions/refinement proofs can be applied in other areas of mathematics, such as graph theory and algebraic topology. They can also be used in computer science and engineering to analyze and simplify complex systems.

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