Edit: Overlapped.
OK folks, after overnight reflection, I concede
Qfwfq's point: neighbourhoods need not be open. The reasons for my conversion will become clear in due course.
So anyway, we now enter the heart of topology. First this;
As always, suppose
is a topological space. A family
of subsets of
is called a
cover for
if
coincides with the union of all
. Clearly any space has at least one cover of some sort. And if the number of
is finite (it may not be), then this is called a finite cover.
If each
is open, it's called an open cover. And if there is a sub-family of
that also covers
, it is called a
sub-cover, naturally enough.
Now, if a finite cover admits of a sub-cover, then this must always be finite. If a cover is
not finite, then any sub-cover may or may not be finite.
So.
If
every open cover of
has a
finite sub-cover, one says that
is
compact.
This is the closest topology allows us to get to the notion of a finite space.
That is, every subset of a compact space is of necessity compact. The converse may or may not be true. A famous example is given by Heine & Borel for
, which I will state later.........
