Exploring the Difference Between Complete and Sequentially Compact Spaces

In summary, the conversation discusses the difference between a complete metric space and a sequentially compact metric space. The main distinction is that in a complete metric space, every Cauchy sequence converges within the space, while in a sequentially compact metric space, every sequence contains a convergent subsequence. The conversation also touches on the definition of a sequence and how it applies to these types of spaces. Ultimately, the properties of these two types of spaces are different and a simple example is given to illustrate this difference.
  • #1
Hymne
89
1
Hello Physicsforums!
I have a problem with the difference between complete metric space and a sequentially compact metric space.
For the first one every Cauchy sequence converges inside the space, which is no problem.
But for the last one "every sequence has a convergent subsequence." (-Wiki) And it's here that I get lost.

How does this affect the constraints on the space?
Could someone please try to give me an intuitive explanation?

For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?
Have I missunderstood it all?
 
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  • #2
Hymne said:
For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?
Have I missunderstood it all?

What is the definition of "sequence"?
 
  • #3
George Jones said:
What is the definition of "sequence"?

Hmm, I use this one http://en.wikipedia.org/wiki/Sequence .
With
In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.
Maybe it´s here that I am confused. :rolleyes:

Should we only work with Cauchy sequences maybe?
 
  • #4
These definitions apply to infinite sequences. (1,2,3,4,5,6) is not an infinite sequence. It doesn't even mean anything for a finite sequence to converge!
 
  • #5
To the original question..

In a complete metric space (an) converges <-> (an) is cauchy

In a compact metric space, every sequence an contains a convergent subsequence (ank).

We should note that convergence -> cauchy in any metric space.

Then, in a compact metric space, every sequence an contains a cauchy subsequence (ank).

Regardless, the properties of these two types of spaces are completely different.

A simple example highlighting the difference between the two is a subset of R1. Consider, the interval (0,1).

By the Heine-Borel theorem, this space is not compact since it is not closed.

It is, however, a complete metric space since cauchy <-> convergent in R1.

Was this your question?
 

FAQ: Exploring the Difference Between Complete and Sequentially Compact Spaces

What is a sequentially compact space?

A sequentially compact space is a mathematical concept used in topology, which is a branch of mathematics that studies the properties of spaces and their relationships. It refers to a topological space where every sequence of elements has a convergent subsequence.

How is sequential compactness different from compactness?

Sequential compactness is a weaker condition than compactness. In a sequentially compact space, every sequence has a convergent subsequence, whereas in a compact space, every open cover has a finite subcover. This means that all compact spaces are sequentially compact, but not all sequentially compact spaces are compact.

What are some examples of sequentially compact spaces?

Some examples of sequentially compact spaces include closed intervals on the real line, the Cantor set, and the unit sphere in finite-dimensional Euclidean space. In general, any compact metric space is also sequentially compact.

How is sequential compactness useful in mathematics?

Sequential compactness is a useful concept in mathematics because it allows us to prove the existence of convergent subsequences in certain spaces. It also plays a key role in the proof of important theorems, such as the Bolzano-Weierstrass theorem and the Heine-Borel theorem.

Are there any real-world applications of sequential compactness?

Sequential compactness has applications in various branches of mathematics, including functional analysis, dynamical systems, and differential equations. It is also used in physics and engineering to study systems that exhibit sequential behavior, such as particle interactions and signal processing.

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