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Bachelier
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Can you provide one to show a separable complete boundd metr. space X is not always seq. compact.
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micromass said:What did you try already? What metric spaces do you know?
micromass said:Yes, that is very good! Try [tex]\mathbb{N}[/tex] with the discrete metric. Isn't that the counterexample you're looking for?
micromass said:You have several (non-equivalent) definitions for dense. The one you mention is dense for ordered sets. However, what we need here is topological dense. Then the definitoin states:
A set D is dense in X if the closure of D is X (or equivalently, that every non-empty open set in X contains a point of D).
With that definition, it can be easily checked that N is indeed dense in itself, and thus separable!
Bachelier said:Can you provide one to show a separable complete boundd metr. space X is not always seq. compact.
Bachelier said:BTW the converse of the question is true. If X is seq. cmpact it is separ. bounded and complete. Right?
In mathematics, a counter example is an example that disproves a statement or a conjecture. In other words, it is an example that contradicts the given statement and shows that it is not universally true.
A Sequentially Compact question is a question that involves a set of numbers or points that are arranged in a specific order, and asks whether this set has a finite subcover. In other words, it asks whether every sequence in the set has a convergent subsequence.
A question can have a counter example to being sequentially compact if there exists a sequence in the set that does not have a convergent subsequence. This would disprove the statement that every sequence in the set has a convergent subsequence, thus showing that the set is not sequentially compact.
Yes, a counter example can be used to prove that the opposite statement is true. If a counter example disproves a statement, then the opposite statement must be true. In the case of a Sequentially Compact question, if a counter example shows that the set does not have a finite subcover, then the opposite statement that the set is not sequentially compact must be true.
Yes, there are several advantages to using a counter example in mathematics. It can help to disprove incorrect statements or conjectures, and also provide insights into the properties and limitations of a particular concept or theorem. It can also lead to the development of new theories and ideas in mathematics. Additionally, using counter examples can improve critical thinking skills and spur further exploration and investigation in a particular area of mathematics.