What are the differences between compact and non compact spaces in topology?

In topology, a compact space is a space in which every open cover has a finite subcover. In other words, a compact space is a space in which every collection of open sets that covers the space can be reduced to a finite collection of open sets that still covers the space.

Non compact spaces are spaces that are not compact. In other words, non compact spaces are spaces in which there exists an open cover that does not have a finite subcover.

Here are some of the key differences between compact and non compact spaces:

  • Compact spaces are sequentially compact. This means that every sequence in a compact space has a convergent subsequence.
  • Non compact spaces are not necessarily sequentially compact. There are non compact spaces in which there exists a sequence that does not have a convergent subsequence.
  • Compact spaces are locally compact. This means that every point in a compact space has a neighborhood that is compact.
  • Non compact spaces are not necessarily locally compact. There are non compact spaces in which there exists a point that does not have a compact neighborhood.

FAQs

Is every compact space closed? Yes, every compact space is closed.

Is every non compact space open? No, non compact spaces are not necessarily open.

Can a space be both compact and non compact? No, a space cannot be both compact and non compact.

Are all finite spaces compact? Yes, all finite spaces are compact.

Are all infinite spaces non compact? No, not all infinite spaces are non compact.

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