Elementary point-set topology today, on a metric space .
- is compact if every open cover of has a finite subcover. One could just as well say “every sequence has a convergent subsequence”, but it’s considered bad form.
- is totally bounded if for every the space can be covered by finitely many balls of radius . One could just as well say “every sequence has a Cauchy subsequence”, but it’s probably bad form, too.
- is complete if every Cauchy sequence converges. I don’t know if anyone considers this sequential definition bad form.
A look at the sequential forms of definitions reveals that a space is compact iff it is complete and totally bounded.
Which of the above properties survive under continuous maps?
- If is compact, then every continuous image of is also compact.
- If is totally bounded, its continuous image may fail to be totally bounded, or even bounded. For example, the interval is continuously mapped onto by .
- if is complete, its continuous image may fail to be complete. For example, the interval is continuously mapped onto by .
There must be other natural examples of how two properties and are not invariant (under some class of transformations) on their own, but are invariant together. None come to mind at this moment.