My thanks to
Qfwfq for answering these queries. But let me emphasize this: the cardinality (i.e. number of elements in) of the powerset
will be
when the cardinality of
is
, but the
topology on
will have cardinality
.
So let's see what we have so far: a set from which we formed all possible subsets, and labelled them as open, closed, both or neither. What's the point of that? Well, none really, so, to follow
maddog, we need to give some additional structure.
But first I have to explain a bit about what it means for a set to be open or closed.
No, even firster, I have to give a familiar example. Consider the real numbers
as a
set. Now sets are just an unordered jumble of things, just like the things in my garage; here they are called real numbers.
But it is an axiom, unprovable as far as I am aware, that the reals are a
total order, that is, for any pair of elements
then
(roughly speaking). Then by our axioms for an allowable topology on
we must have that the union of arbitrary open subsets of
is itself open.
Standard terminology calls the open sets in
as
, and the union of all such open sets will thus be the open set
. This union is called the (topological) real line
, which must be open since
isn't "really" a real number. This is called the
standard topology on the reals.
OK? Oh, and if you have trouble convincing yourself that
are open, I suggest you follow
Qfwqf's prof's advice and take up drinking!
Let's adopt a perfectly standard abuse of notation: since we almost never care which particular topology we are talking about, nobody, but
nobody uses
to denote a topological space. They simply say "Let
be a topological space".
Let's now see what it means for a set to be open or closed. Let
be a subset (open or closed) of a topological space. Define the
interior of
as the largest open set contained
in , and write this as
. Then quite clearly, if
then
is open. ("largest" here means the union of all open sets entirely within the subset)
Now define the
closure of
as the smallest closed set that
contains . Write this as
. Obviously, if
then
is closed. (And "smallest" here means the intersection of all closed sets that completely contain the subset)
Finally define the
boundary of
as
(where the back-slash is the set theoretic version of arithmetic "minus".
Now you're going to ask what is the boundary of a set that is both open and closed? What if it is neither? Ah, wait!
I trust the suspense won't kill you.....
