Exploring the Paradox of Infinite Subcovers in Compact Spaces

In summary, a compact set is defined as one where every open cover is either finite or has a finite sub-cover. This means that even if we start with an open cover C1 and find a finite sub-cover C2, we can continue finding finite sub-covers like C3, Cz, etc. However, there may be a confusion with the symbol \subset, which can mean either a subset or a proper subset. This ambiguity can also be seen in the definition of compactness, as some older texts use the symbol in its more general meaning. Ultimately, this confusion may stem from a desire to appear more sophisticated than to promote clarity in mathematics.
  • #1
quantum123
306
1
According to definition, a compact set is one where every open cover has a finite sub-cover.
So let say I have C1, which is an open cover, I have C2 subset of C1 which is also an open cover. But C2 is finite.
But since C2 is an open cover then there is a finite subcover C3 which is subset of C2.
And so on and so forth , we will definitely end up with Cz which may only have one element. Then there will be no more subset of Cz, then how can there be any more subcover?
Isn't there a contradiction?
 
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  • #2
quantum123 said:
But since C2 is an open cover then there is a finite subcover C3 which is subset of C2.

Yes, but C3 can be the same as C2, it doesn't have to have smaller cardinality.
 
  • #3
I see. So you mean subset but not necessarily proper subset suffices?
I have gotten confused by the symbol.

[tex]
\subset[/tex]

Does it mean subset or proper subset?
 
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  • #4
This was exactly the same difficulty that I had with the notion of a compact set. I originally though a compact set meant a single point! But it doesn't.

quantum123 said:
[tex]
\subset[/tex]

Does it mean subset or proper subset?
In modern notation, it would mean a proper subset, but much of the older texts and definitions of topology use it in its more ambiguous meaning as simply a "subset", proper or equal.

A more straightforward definition of compactness is simply to say that:
A compact set is one where every open cover is either finite or has a finite sub-cover.

A compact set is an extention of the idea of a closed bounded set in spaces where neither close nor bounded makes much sense. In euclidean space, or spaces isomorphic to some euclidean space, compactness is equivilant to being closed and bounded.

I think the old notion of allowing a subset to be less than or equal to is a notion that really should be retired. It's confusing, especially in cases like this. It's continued survival is probably for reasons of ostentation rather than clarity.
 
  • #5
"A compact set is one where every open cover is either finite or has a finite sub-cover." - Good definition!

That is one cursed problem in math for the new-comer.
The other one is the

[tex]
\theta
[/tex]
[tex]
\phi
[/tex]


Different authors define it differently.
I mean maths is supposed to be clear, logical and universal, how can that happen?
 
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  • #6
quantum123 said:
I mean maths is supposed to be clear, logical and universal, how can that happen?
Pretension.
 

1. What is a compact space?

A compact space is a mathematical concept used to describe a topological space that is "small" in some sense. Intuitively, a compact space is one that can be covered by a finite number of sets, each of which are "small" in some way.

2. How is compactness different from other mathematical concepts?

Compactness is a property that is specific to topological spaces, and is different from other mathematical concepts such as openness, connectedness, and continuity. While these concepts describe specific aspects of a space, compactness is a more general property that encompasses all of these concepts and more.

3. What are some examples of compact spaces?

Some common examples of compact spaces include a closed interval on the real line, a finite set, and the surface of a sphere. Other examples include the Cantor set, the Sierpinski triangle, and the Menger sponge.

4. What are the benefits of studying compact spaces?

Compact spaces have many important applications in various fields of mathematics, including topology, analysis, and geometry. They also provide a useful tool for simplifying and understanding complex spaces, as well as for proving theorems and solving problems.

5. How can the concept of compactness be applied in real-world situations?

Compactness has many real-world applications, such as in computer science, where it is used to optimize algorithms and data structures. It is also used in economics and game theory to model decision-making processes. Additionally, compactness can be applied in engineering and physics to study the behavior of physical systems.

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