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Imagine you have a collection of blankets that are large enough to entirely cover you when you're chilly. In mathematics, we use a similar concept for sets, which is called an open cover. More precisely, an open cover of a set in a topological space is a collection of open sets whose union completely encloses the set in question.

For instance, let's consider a set, akin to how you'd be lying on a field. Now, if we take several transparent sheets, each one representing an open set, and lay them over you so that every part of you is covered—that forms what we call an 'open cover'. The beauty is that some sheets may overlap, and you might be covered multiple times, but as long as every part of you is under at least one sheet, we're satisfied that it’s an open cover.

Now let's say we have far too many sheets, and we want to be sustainable—only using what's necessary to keep you covered. What if we could select just a few of these sheets that still do the job? This action refers to a finite subcover. It’s essentially a smaller collection of the open sets from our original 'open cover' that still completely covers our set.

This concept is at the heart of 'compactness'. A set is compact if, whenever it has an open cover, it also possesses a finite subcover that still envelops the entire set. In this context, the compactness property ensures we can always find a 'minimal' cover—much like selecting the fewest possible blankets needed to keep warm. In our textbook exercise, this step was crucial for proving that the union of several compact sets maintained the desirable trait of being compact.

When the winter is harsh, sometimes one blanket isn't enough, so we gather more. Similarly, sets can be bundled together to form a bigger set. The exercise you encountered focuses on the concept of the union of compact sets.

This idea is like taking several 'warm' and compact 'blankets' (compact sets) and tossing them together to form a bigger one. As each compact blanket can be covered by a small number of sheets (finite subcover), when you combine all these separate layers into a bigger blanket, you still end up with a 'warm' bundle that can be covered by a finite combination of your original sheets.

This property is vital in topology and analysis as it underpins the behavior of functions and spaces in more complex scenarios. By showing that the combination of a finite number of compact sets is still compact, we ensure that problems can be 'localized' to manageably small pieces, just like using several small blankets efficiently rather than one massive one.