How can you tell if a specific topological space is compact?

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Discussion Overview

The discussion revolves around the concept of compactness in topological spaces, exploring how to determine if a specific topological space is compact. It includes theoretical aspects, definitions, and theorems related to compactness.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant notes the difficulty of determining compactness directly from its definition.
  • Another participant mentions several theorems that assist in identifying compact spaces, such as the product of compact spaces being compact and that closed subspaces of compact spaces are also compact.
  • A participant seeks clarification on the notion of a cover, specifically whether a set A can be a proper subset of its cover when discussing compactness.
  • It is proposed that if A is equal to the union of the cover, A would be open, which is typically associated with being "not compact."
  • One participant expresses skepticism about learning from the internet compared to traditional library resources.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and approaches to the concept of compactness, with no consensus reached on the best methods for determining compactness or the value of different learning resources.

Contextual Notes

There are unresolved assumptions regarding the definitions of covers and compactness, and the discussion does not clarify the implications of subsets versus proper subsets in the context of compactness.

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eddo's thread got me thinking: How can you tell if a specific topological space is compact? It seems like it would be hard to do just starting with the definition of compactness.
 
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Yes, it would, that is why we have lots of theorems to help: the product of compact spaces is compact. Any closed subspace of a compact space is compact. Every compact metric space is sequentially compact. Usually, though I think it would be easier to show something wasn't compact by finding an explicit open subcover without refinement. Plus we often know a lot about the topological space other than simply that it is a topological space.
 
Thanks, I'll look into those theorems you listed. Also, I wanted to make sure I have the notion of a cover straight. If a family of sets form a cover for a set A, then that means A is contained in the union of those sets, right? The reason I ask is that on the Wolfram Mathworld site, they say
A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose union is X.
In this case, we are talking about the whole topological space X, so is that why they say X is the union of a family of open sets? I just want to make sure that when talking about the compactness of a subset A of X, it is okay for A to be a proper subset of its cover, rather than equal to it.
 
If A is a subset of a topological space, then it is almost certain that A must be a proper subset of the cover, otherwise A if it were equal to the union of the cover would be open, which is usually a way to be "not compact".
 
Why don't people just go to the library and read books instead of trying to learn it from the internet? One cannot learn a subject by reading encyclopedia entries.
 
matt:
Thanks for clearing that up!


mruncleramos said:
Why don't people just go to the library and read books instead of trying to learn it from the internet? One cannot learn a subject by reading encyclopedia entries.
Who is you? Where you comin' from? :cool:
 

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