Send a compact space continuously onto another and the target is compact too. Cover the target, pull the cover back, thin it where compactness lives, and push the thinned cover forward.

A tour guide who can see the whole group at once from a few vantage points. Wherever the group is led, a few vantage points still suffice, because each one came from a place it was already known to work.

Let $f$ be continuous from $X$ to $Y$ and onto, meaning every point of $Y$ is $f(x)$ for some $x$. If $X$ is compact then so is $Y$. A map that is not onto is covered by the same theorem: apply it to the image with the topology the previous chapter gave it, and the conclusion reads that the image is compact.

• Continuity is used once, to turn the cover of $Y$ into a cover of $X$ by pulling it back.
• Compactness of $X$ is used once, to thin that pulled-back cover to finitely many.
• Being onto is used once, at the end, to know that every point of $Y$ is a value and so is reached by one of the chosen members.
• Nothing is claimed in the other direction: a compact image says nothing about the space it came from, since every space maps onto a single point.

Members of an open cover of $Y$, and the open sets of $X$ they pull back to, $U_i = f^{-1}(V_i)$. Pulling a cover back gives a cover, since every point of is sent somewhere and that somewhere lies in some . Compactness of then thins the pullback to finitely many, and the matching cover because is onto. The ovals name the members rather than fill the box — a cover reaches every point, and no oval reaches a corner.

Take any open cover of the target. Pull each member back through the map; continuity says each pullback is open, and together they cover the source, because every point of the source is sent somewhere and that somewhere lies in one of the members. Now the source is compact, so finitely many of the pullbacks already cover it. Those finitely many came from finitely many members of the original cover, and those members cover the target: any point there is a value of the map, the point it came from lies in one of the chosen pullbacks, and so the value lies in the matching member.

Take any open cover of the target; the definition has to be checked for it.

The map is continuous, so every one of these pullbacks is open in the source.

Every point of the source is sent to a point of the target, which lies in some member, so the pullbacks cover.

The source is compact, so finitely many of the pullbacks already cover it.

Every point of the target is a value; the point it came from lies in one of the chosen pullbacks, so the value lies in the matching member.

The one step continuity is used for.

The one step that needs the map to be onto.

Continuous, onto, compact image, and the source is not compact.

The step compactness of the source provides.