Proof of Subspace Topology Problem: Error Identification & Explanation

In summary, the conversation discusses a mathematical proof and its flaws. The speaker points out that the proof relies on the assumption that a certain set is closed when it has not been proven to be so. They also note that the proof does not refer to being open in a specific set, which is necessary for the proof to hold. The speaker suggests looking at the definition of subspace topology to understand why the statement holds.
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Homework Statement
Let [itex]K[/itex] be a subset of a metric space [itex]M[/itex], then if [itex]U[/itex] is open, [itex]U\cap K[/itex] is open in [itex]K[/itex]
Relevant Equations
Closed set: The set is called closed if all convergent sequences of elements of the set converge to an element of the set.

Open set: A set [itex]X[/itex] is said to be open if for every point [itex]x[/itex] in the set there is an open ball centered in it that is contained in the set.

Closure: The closure of a set [itex]A[/itex] is the intersection of all closed sets that contain [itex]A[/itex]
I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point [itex]x[/itex] of [itex]U[/itex] that is not in [itex]U[/itex], but is in [itex]K[/itex] (in other words [itex]x \in K \cap(\overline{U}-U)[/itex]), then suppose that [itex]K\cap U[/itex] is closed, this implies [itex]\overline{K \cap U}=K \cap U[/itex] and then must contain all limit points of [itex]K\cap U[/itex] since [itex]x[/itex] is a limit point of [itex]U[/itex] and is in [itex]K[/itex], it is also a limit point of [itex]K\cap U[/itex] and therefore must be in it since its closed. However, this is absurd since it would imply that [itex]x \in U[/itex] then [itex]K \cap U[/itex] must be open.
 
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  • #2
You appear to have assumed ##K\cap U## is closed and tried to derive a contradiction from that. If you have successfully done that (I didn't check) then you can conclude that ##K\cap U## is not closed. However that does not imply that it is open. Many sets are neither open nor closed, eg the half-open interval [0,1).

Also, you have not at any point in your proof attempt referrred to being open in K, which is what you have to prove. Being open in K is different from just being open (ie open in M). In a problem like this you need to make clear, when you refer to a set being open or closed, whether you mean open or closed in K or in M.
 
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  • #3
If you look at the definition of subspace topology, and what the open sets are specifically of K, it will be immediate why the statement holds.
 
  • #4
nucl34rgg said:
If you look at the definition of subspace topology, and what the open sets are specifically of K, it will be immediate why the statement holds.

I think this is probably not true. In particular K is a metric space, and so has a topology from that, which you are probably supposed to use. I suspect this question is leading up to motivating why the definition of the subspace topology makes sense.
 

1. What is the Proof of Subspace Topology Problem?

The Proof of Subspace Topology Problem is a mathematical concept that involves identifying and explaining errors in a given subspace topology. It is commonly used in topology and functional analysis to analyze the properties of topological spaces.

2. How do you identify errors in a subspace topology?

To identify errors in a subspace topology, one must first understand the definition and properties of a subspace. Then, the researcher must carefully analyze the given subspace topology and compare it to the properties of a subspace. Any discrepancies or inconsistencies between the two will indicate errors in the subspace topology.

3. What is the importance of identifying errors in a subspace topology?

Identifying errors in a subspace topology is crucial in ensuring the accuracy and validity of mathematical proofs and analyses. It allows for a deeper understanding of the properties of a topological space and can lead to the discovery of new theorems and concepts.

4. Can you provide an example of a subspace topology problem and its error identification?

Sure, for example, let's say we have a subspace topology that claims to be Hausdorff, but upon closer inspection, we notice that there is a sequence of points that converges to two distinct points. This contradicts the definition of a Hausdorff space, which states that any two distinct points must have disjoint neighborhoods. Therefore, the error in this subspace topology would be the claim of being Hausdorff.

5. How can one explain the identified errors in a subspace topology?

To explain the identified errors in a subspace topology, one must provide a detailed analysis of the properties and definitions of a subspace and compare them to the given subspace topology. This explanation should clearly state the discrepancies and inconsistencies between the two and how they affect the validity of the subspace topology.

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