How Can Infinite Subcovers Exist in Compact Spaces Without Contradiction?

  • Context: Graduate 
  • Thread starter Thread starter quantum123
  • Start date Start date
  • Tags Tags
    Compact Space
Click For Summary

Discussion Overview

The discussion revolves around the concept of compact sets in topology, specifically addressing the existence of infinite subcovers in compact spaces and the potential contradictions that arise from the definitions and interpretations of open covers and subcovers.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant asserts that a compact set is defined such that every open cover has a finite subcover, leading to confusion about the nature of subsets and whether they can be proper or not.
  • Another participant clarifies that a finite subcover does not necessarily have to be a proper subset, suggesting that the same set can serve as both the cover and the subcover.
  • There is a discussion about the notation for subsets, with some participants expressing confusion over whether the symbol \subset denotes a subset or a proper subset.
  • A participant shares their initial misunderstanding of compact sets, indicating that they thought it referred to a single point, highlighting the complexity of the concept.
  • One participant proposes a more straightforward definition of compactness, emphasizing that it should be clear and unambiguous, while others express frustration over the varying definitions found in different texts.
  • Concerns are raised about the clarity and universality of mathematical definitions, with one participant commenting on the pretentiousness that can arise in mathematical discourse.

Areas of Agreement / Disagreement

Participants express varying levels of confusion and disagreement regarding the definitions and implications of compact sets and open covers. There is no consensus on the clarity of the definitions or the appropriateness of certain notations.

Contextual Notes

Participants note the ambiguity in the definition of subsets and the historical context of mathematical notation, which may contribute to misunderstandings. The discussion reflects a range of interpretations and assumptions about compactness and its implications.

quantum123
Messages
306
Reaction score
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?
 
Physics news on Phys.org
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.
 
I see. So you mean subset but not necessarily proper subset suffices?
I have gotten confused by the symbol.

<br /> \subset

Does it mean subset or proper subset?
 
Last edited:
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:
<br /> \subset

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.
 
"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

<br /> \theta<br />
<br /> \phi<br />


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

Similar threads

Graduate Understanding: double of conformally flat manifold is conformally flat
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Another similar notion to compactness
  • · Replies 2 ·
Replies
2
Views
558
A proof that a union of compact spaces is compact
  • · Replies 12 ·
Replies
12
Views
3K
Graduate Proving a product of compact spaces is compact
  • · Replies 2 ·
Replies
2
Views
8K
Graduate Compact sets in Hausdorff space are closed
  • · Replies 4 ·
Replies
4
Views
5K
Graduate Compact Sets: Need help understanding
  • · Replies 2 ·
Replies
2
Views
2K
Graduate What subset of a 3D space can be addressed by a single chart?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad On two definitions of locally compact
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Understanding Compactness in ##R^2##: Analyzing Boundaries and Open Disks
  • · Replies 12 ·
Replies
12
Views
2K
Graduate Confusion about definition of compactness
  • · Replies 8 ·
Replies
8
Views
3K