Strength in unity

Elementary point-set topology today, on a metric space X.

  • X is compact if every open cover of X has a finite subcover. One could just as well say “every sequence has a convergent subsequence”, but it’s considered bad form.
  • X is totally bounded if for every r>0 the space can be covered by finitely many balls of radius r. One could just as well say “every sequence has a Cauchy subsequence”, but it’s probably bad form, too.
  • X 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 X is compact, then every continuous image of X is also compact.
  • If X is totally bounded, its continuous image may fail to be totally bounded, or even bounded. For example, the interval (0,1] is continuously mapped onto [1,\infty) by x\mapsto x^{-1}.
  • if X is complete, its continuous image may fail to be complete. For example, the interval [1,\infty) is continuously mapped onto (0,1] by x\mapsto x^{-1}.

There must be other natural examples of how two properties P_1 and P_2 are not invariant (under some class of transformations) on their own, but are invariant together. None come to mind at this moment.

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