4Compactness
IB Metric and Topological Spaces
4 Compactness
Compactness is an important concept in topology. It can be viewed as a
generalization of being “closed and bounded” in
R
. Alternatively, it can also be
viewed as a generalization of being finite. Compact sets tend to have a lot of
really nice properties. For example, if
X
is compact and
f
:
X → R
is continuous,
then f is bounded and attains its bound.
There are two different definitions of compactness - one based on open covers
(which we will come to shortly), and the other based on sequences. In metric
spaces, these two definitions are equal. However, in general topological spaces,
these notions can be different. The first is just known as “compactness” and the
second is known as “sequential compactness”.
The actual definition of compactness is rather weird and unintuitive, since it
is an idea we haven’t seen before. To quote Qiaochu Yuan’s math.stackexchange
answer (http://math.stackexchange.com/a/371949),
The following story may or may not be helpful. Suppose you live
in a world where there are two types of animals: Foos, which are
red and short, and Bars, which are blue and tall. Naturally, in your
language, the word for Foo over time has come to refer to things
which are red and short, and the word for Bar over time has come to
refer to things which are blue and tall. (Your language doesn’t have
separate words for red, short, blue, and tall.)
One day a friend of yours tells you excitedly that he has discovered
a new animal. “What is it like?” you ask him. He says, “well, it’s
sort of Foo, but. . . ”
The reason he says it’s sort of Foo is that it’s short. However,
it’s not red. But your language doesn’t yet have a word for “short,”
so he has to introduce a new word — maybe “compact”. . .
The situation with compactness is sort of like the above. It
turns out that finiteness, which you think of as one concept (in the
same way that you think of “Foo” as one concept above), is really
two concepts: discreteness and compactness. You’ve never seen
these concepts separated before, though. When people say that
compactness is like finiteness, they mean that compactness captures
part of what it means to be finite in the same way that shortness
captures part of what it means to be Foo.
But in some sense you’ve never encountered the notion of com-
pactness by itself before, isolated from the notion of discreteness (in
the same way that above you’ve never encountered the notion of
shortness by itself before, isolated from the notion of redness). This
is just a new concept and you will to some extent just have to deal
with it on its own terms until you get comfortable with it.