other · 2017
Compact Sets
Daniel J. Madden, Jason A. Aubrey
Cited by 0Abstract and details only
Abstract
A set that is closed and bounded has many more strong properties than a set that is just closed or just bounded. Proving many of the properties of these closed and bounded sets requires greater logical power than directly applying the two definitions separately. Mathematicians have developed an amazing and powerful trick that provides a completely different path to such proofs. A theorem is proved in the chapter that says that, for any closed and bounded set, a very nonintuitive trick works in a proof. Compact sets are simply sets that are closed and bounded. The proof of this is long and detailed. When one assumes that a set is compact (or closed and bounded after Heine-Borel is proved), it is often a good idea not to write out the definition of compact for that set. Writing out that definition can result into tricking ourselves into thinking that he/she has a cover of the set.
Cite this paper
Madden, D. J., & Aubrey, J. A. (2017). Compact Sets. https://doi.org/10.1002/9781119314769.ch18
Madden, Daniel J., and Jason A. Aubrey. “Compact Sets.” 2017, pp. 345–57, https://doi.org/10.1002/9781119314769.ch18.
Madden, Daniel J., and Jason A. Aubrey. 2017. “Compact Sets.” https://doi.org/10.1002/9781119314769.ch18.
Madden, D.J. and Aubrey, J.A. (2017) ‘Compact Sets’. Available at: https://doi.org/10.1002/9781119314769.ch18.
@misc{madden2017compact,
title = {{Compact Sets}},
author = {Madden, Daniel J. and Aubrey, Jason A.},
pages = {345--357},
year = {2017},
doi = {10.1002/9781119314769.ch18}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.