Metric Spaces & Compactness - Apostol Theorem 4.28

In summary, the proof of Theorem 4.28 in Tom Apostol's book Mathematical Analysis (2nd Edition) explains that if m = inf f(X), then there exists a sequence of points in X whose limit is m. This implies that m is adherent to f(X), meaning that it is in the closure of f(X). This concept is further clarified with the definitions of adherent and accumulation points.
  • #1
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I need help with the proof of Theorem 4.28 in Tom Apostol's book: Mathematical Analysis (2nd Edition).

Theorem 4.28 reads as follows:View attachment 3855In the proof of the above theorem, Apostol writes:

" ... ... Let \(\displaystyle m = \text{ inf } f(X)\). Then \(\displaystyle m\) is adherent to \(\displaystyle f(X)\) ... ... "

Can someone please explain to me exactly why \(\displaystyle m = \text{ inf } f(X)\) implies that \(\displaystyle m\) is adherent to \(\displaystyle f(X)\)?

Help will be appreciated ... ...NOTE: In the above proof Apostol makes mention of an adherent point, so I am providing Apostol's definition of an adherent point together with (for good measure) his definition of an accumulation point ... ...View attachment 3856
 
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  • #2
Hi Peter,

$m=inf \ f(X)$ implies that there exists a sequence $\{x_{n}\}_{n\in \Bbb{N}}\subseteq X$ such that $\underset{n \to +\infty}{lim}f(x_{n})=m$,
so $m$ is in the closure of $f(X)$
 
  • #3
Fallen Angel said:
Hi Peter,

$m=inf \ f(X)$ implies that there exists a sequence $\{x_{n}\}_{n\in \Bbb{N}}\subseteq X$ such that $\underset{n \to +\infty}{lim}f(x_{n})=m$,
so $m$ is in the closure of $f(X)$
Thanks for the help Fallen Angel ...

I guess for the case of finite sets such as \(\displaystyle A \subset \mathbb{R}\) where \(\displaystyle A = \{1,2,3 \}\) and \(\displaystyle inf A = 1\), the sequence \(\displaystyle \{x_n \}\) would be \(\displaystyle 1,1,1, \ ... \ ... \ \)

Is that correct?

Peter
 
  • #4
Hi Peter,

Yes, it's correct, but this sequences are not unique, $\{2,2,3,2,3,1,1,1,1,\ldots\}$ it's another one, for example.
 
  • #5


An adherent point of a set A is a point x such that every neighborhood of x contains at least one point of A.

An accumulation point of a set A is a point x such that every neighborhood of x contains infinitely many points of A.

To understand why m = inf f(X) implies that m is adherent to f(X), we first need to understand what inf f(X) represents. In this case, inf f(X) represents the infimum (greatest lower bound) of the set of values of f(X). This means that m is the smallest value that f(X) can take on.

Now, let's consider what it means for m to be adherent to f(X). This means that every neighborhood of m contains at least one point of f(X).

Since m is the smallest value that f(X) can take on, this means that every neighborhood of m contains at least one point of f(X). In other words, every neighborhood of m contains a value of f(X) that is smaller than or equal to m.

But according to the definition of infimum, m is the smallest value that f(X) can take on. This means that there are no values of f(X) that are smaller than m. Therefore, every neighborhood of m must contain at least one point of f(X) that is equal to m.

Thus, m = inf f(X) implies that m is adherent to f(X) because every neighborhood of m contains at least one point of f(X), and there are no values of f(X) that are smaller than m.
 

FAQ: Metric Spaces & Compactness - Apostol Theorem 4.28

1. What is a metric space?

A metric space is a mathematical concept that consists of a set of objects and a function called a metric, which measures the distance between any two objects in the set. The metric must follow certain properties, such as being non-negative, symmetric, and satisfying the triangle inequality.

2. What is compactness?

Compactness is a property of a metric space that indicates that the space is "small" or "finite" in some sense. It means that every sequence of points in the space has a convergent subsequence, meaning it approaches a specific point in the space. In simpler terms, compactness means that a space does not have any "holes" or "gaps" in it.

3. What is Apostol Theorem 4.28?

Apostol Theorem 4.28, also known as the Heine-Borel Theorem, states that a metric space is compact if and only if it is closed and bounded. This means that a space must contain all of its boundary points and have a finite size in order to be considered compact.

4. How is Apostol Theorem 4.28 useful?

Apostol Theorem 4.28 is useful in many areas of mathematics, such as analysis, topology, and geometry. It provides a way to prove the compactness of a space, which is a fundamental concept in many mathematical theories and applications. It also allows for the simplification of proofs in various mathematical theorems.

5. Can Apostol Theorem 4.28 be applied to any metric space?

No, Apostol Theorem 4.28 can only be applied to metric spaces that satisfy the properties of compactness, such as being closed and bounded. It cannot be applied to spaces that do not have a well-defined metric or do not follow the necessary properties. Additionally, the theorem only applies to metric spaces and not other types of mathematical spaces, such as topological spaces.

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