Nowhere dense subset of a metric space

In summary, a nowhere dense subset of a metric space can be not closed if it contains a Cauchy sequence without its limit point, such as 1/2^n with a limit point of 0. In a closed subset of a metric space, every limit point is contained in the subset.
  • #1
de_brook
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Can we have some examples in which a nowhere dense subset of a metric space is not closed?
 
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  • #2
de_brook said:
Can we have some examples in which a nowhere dense subset of a metric space is not closed?

just take a Cauchy sequence without its limit point e.g. 1/2^n
 
  • #3
Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero?
 
  • #4
"Is zero not a limit point of 1/2^n since as n gets large, 1/2^n goes to zero? "

Yes, and that is precisely the issue here. A closed subset of a metric space

(I think this is true in any topological space)contains all its limit points. One

way of seeing this is seeing what would happen if the limit point L of a closed

set C in X was not contained in C. Then L is in X-C, and every 'hood (neighborhood)

of L in X-C , intersects points of C.
 
  • #5


A nowhere dense subset of a metric space is a subset that has no interior points, meaning that every point in the subset is isolated and has no other points nearby. This can happen in a variety of metric spaces, such as Euclidean spaces, Banach spaces, and metric graphs.

One example of a nowhere dense subset in a metric space that is not closed is the set of rational numbers in the interval [0,1]. This set is nowhere dense because for any given point in the interval, there will always be irrational numbers infinitely close to it. However, this subset is not closed because it does not contain all of its limit points, such as the irrational numbers.

Another example is the set of algebraic numbers in the real numbers. This set is nowhere dense because for any given point, there will always be transcendental numbers infinitely close to it. However, it is not closed because it does not contain all of its limit points, such as the irrational numbers.

In general, any subset in a metric space that is not closed will also be nowhere dense because it will have points that are isolated and not part of the subset. Therefore, a nowhere dense subset can only be closed if it contains all of its limit points.
 

What is a nowhere dense subset of a metric space?

A nowhere dense subset of a metric space is a subset of the metric space that has no interior points and is not dense in any part of the space. This means that for any point in the metric space, there exists an open ball around that point that does not contain any points from the subset.

What is the importance of studying nowhere dense subsets in a metric space?

Nowhere dense subsets play a significant role in the study of topological properties of metric spaces. They help to define concepts such as separability and completeness, and can also be used to characterize certain types of spaces, such as Banach spaces and Fréchet spaces.

How are nowhere dense subsets related to dense subsets?

A subset of a metric space is considered dense if every point in the space is either an interior point of the subset or a limit point of the subset. Nowhere dense subsets are the complements of dense subsets, meaning that they contain all the points that are not in the dense subset.

Can a metric space have more than one nowhere dense subset?

Yes, a metric space can have multiple nowhere dense subsets. In fact, every metric space has at least one nowhere dense subset, which is the empty set. Other examples of nowhere dense subsets include finite sets, sets of isolated points, and Cantor sets.

How can one prove that a subset is nowhere dense in a metric space?

To prove that a subset is nowhere dense in a metric space, one can use the definition of nowhere dense to show that for any point in the metric space, there exists an open ball around that point that does not contain any points from the subset. This can be done by assuming the contrary and arriving at a contradiction.

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