Understanding Compactness in Metric Spaces: Closed, Bounded, and Open Covers

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In summary, the conversation discusses the concept of compactness in a metric space. It is mentioned that a set being closed and bounded does not always guarantee that it is compact, and an example is given using the real numbers. The question arises about choosing an open cover and whether it should be relative to the larger metric space or the subspace. It is clarified that metric spaces are not typically viewed as being in a larger metric space.
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Buri
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My professor in lecture yesterday said that if a set is closed and bounded in a metric space it doesn't necessarily imply that it is compact. If X = R^n, then it does happen to be true, however. I was trying to construct an example, but I am getting confused. If I let X = R, and Y = (0,1) where Y is a subspace of X, then A = (0,1/2] is closed and bounded in Y. However, from where do I choose the open cover? That is, open relative to X or Y? I know in this case it won't make a difference, but maybe in differently chosen X, Y and A it might. I guess this is a matter of definition, but would like some help.

Thanks a lot.
 
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Don't think about a subset of Rn. Metric spaces just are, they aren't normally viewed as being in a larger metric space even if that is useful. In this case you have X=(0,1) with the Euclidean metric, and A=(0,1/2]. The fact that you can embed this metric space into the real numbers is irrelevant
 
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Ahh I see. Thanks!
 

1. What is a metric space?

A metric space is a mathematical concept that describes a set of points with a distance function between them. The distance function, or metric, defines how to measure the distance between any two points in the set.

2. How is compactness defined in a metric space?

In a metric space, compactness is defined as the property of a set where every open cover of the set has a finite subcover. This means that a set is compact if it can be covered by a finite number of open sets.

3. What is the importance of compactness in mathematics?

Compactness is an important concept in mathematics because it allows for the study of infinite sets and their properties. It is also used in various branches of mathematics such as analysis, topology, and geometry.

4. How do you determine if a metric space is compact?

To determine if a metric space is compact, one can use the Heine-Borel theorem, which states that a metric space is compact if and only if it is closed and bounded. Another method is to check if every sequence in the space has a convergent subsequence.

5. Can non-metric spaces be compact?

No, non-metric spaces cannot be compact. The concept of compactness is only defined for metric spaces, which have a defined distance function between points. Other types of spaces, such as topological or normed spaces, have their own definitions of compactness.

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