# 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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