Question

Let \(M\) be a totally bounded subset of a metric space \(R\). Prove that the \varepsilon-nets figuring in the definition of total boundedness of \(\mathrm{M}\) can always be chosen to consist of points of \(\mathrm{M}\) rather than of \(\mathrm{R}\). Hint. Given an \(\varepsilon\)-net for M consisting of points \(a, a, \ldots, a, \in R\), all within \(\varepsilon\) of some point of \(M\), replace each point \(a\), by a point \(b\), EM such that \(\rho\left(a_{k}, b_{k}\right) \leqslant E\).

Short answer

An \(\varepsilon\)-net for \(M\) can be modified to consist only of points from \(M\).

Step-by-step solution

Understanding Total Boundedness

A subset \(M\) of a metric space is totally bounded if for every \(\varepsilon > 0\), there exists a finite \(\varepsilon\)-net for \(M\), meaning a finite set of points from the metric space such that every point of \(M\) is within distance \(\varepsilon\) from some point of this finite set.

Constructing an Existing \(\varepsilon\)-Net

Consider that an \(\varepsilon\)-net for \(M\) is given, consisting of points \(a_1, a_2, \ldots, a_n\) from \(R\). This means each point in \(M\) is within \(\varepsilon\) distance to at least one \(a_i\). However, \(a_i\) may not be in \(M\).

Replacing Points with \(M\)-Points

For each \(a_k\), find a point \(b_k\) in \(M\) such that the distance \(\rho(a_k, b_k) \leq \varepsilon\). This is possible as each \(a_k\) is within \(\varepsilon\) of some point in \(M\).

Forming the New \(\varepsilon\)-Net

The set \(\{b_1, b_2, \ldots, b_n\}\) becomes the \(\varepsilon\)-net consisting of points from \(M\). Every point in \(M\) is within \(\varepsilon\) of a point \(a_k\) and \(\rho(a_k, b_k) \leq \varepsilon\). Therefore, every point in \(M\) is within \(2\varepsilon\) of a point \(b_k\).

Conclusion

We have constructed an \(\varepsilon\)-net consisting solely of points from \(M\). This satisfies the definition of total boundedness but with the requirement that the net points are within \(2\varepsilon\) distance.

Key concepts

Total Boundedness

In the realm of metric spaces, the concept of total boundedness is pivotal in understanding how sets can be covered by small "neighborhoods". A subset \( M \) of a metric space is termed totally bounded if, for every slight positive distance (represented by \( \varepsilon > 0 \)), you can find a finite collection of points such that every point in \( M \) is within a distance \( \varepsilon \) from at least one of these selected points.
This finite collection is known as an \( \varepsilon \)-net. It acts as a kind of safety net where every point in the subset \( M \) is "caught" by being close to one of the points in the net. It's like trying to cover all areas of the set \( M \) with tiny balls of radius \( \varepsilon \) centered around these selected points.
Understanding this concept is crucial because it sets the stage for discussions about compactness. Knowing a space is totally bounded can aid in determining if it is compact when combined with completeness. Total boundedness ensures that even though a set may be infinite, its spread in the metric space is contained within a finite mainframe.

Epsilon-Net

An \( \varepsilon \)-net serves as a finite approximation of a set within a metric space. It provides a structured way of ensuring that every element from the subset \( M \) is close enough to at least one point from a finite selected subset of points.
Imagine you are given a set of points scattered around an area. If you want to build an \( \varepsilon \)-net for this set, you aim to pick specific points such that every point in your original set is no more than \( \varepsilon \) distance away from one of your chosen points.
In our original exercise, even if these chosen points initially don't come from the subset \( M \) but from the larger metric space \( R \), they can be smartly substituted. As the exercise describes, you can replace each point in the existing \( \varepsilon \)-net with another point from \( M \), ensuring the new net remains within a controlled distance from all points in \( M \). This reformed \( \varepsilon \)-net thus guarantees each part of \( M \) is correctly approximated, staying essential for deeper evaluations of the subset.

Totally Bounded Sets

Totally bounded sets have a mysterious yet fascinating nature. They can appear infinite but can be enclosed within a broader metric space by finite means. This makes them an intriguing subject in mathematical discussions, especially in metric spaces.
To visualize a totally bounded set, think of trying to place a finite number of small circles over the entirety of the set, ensuring complete coverage. If such a finite cover is possible for any \( \varepsilon \), no matter how tiny, the set is regarded as totally bounded.
In practical terms, identifying a subset as totally bounded can simplify complex tasks like proving compactness in metric spaces. They frame a clear understanding that the set’s spread is limited, bringing a sense of orderliness to its infinite nature. With this concept, one can transition a myriad of theoretical problems into more manageable studies, often leading to interesting and applicable results in analysis.