Separation Axioms
Now that we have some vocabulary about separation properties down we can talk about properties of spaces as a whole, called the separation axioms.
First off, we say that a space is if every two distinct points can be topologically distinguished. This fails, for example, in the trivial topology on a set
if
has at least two points, because every point has the same collection of neighborhoods —
for all points
. As far as the topology is concerned, all the points are the same. This turns out to be particularly interesting in conjunction with other separation axioms, since we often will have one axiom saying that a property holds for all distinct points, and another saying that the property holds for all topologically distinguishable points. Adding
turns the latter version into the former.
Next, we say that a space is if any two topologically distinguishable points are separated. That is, we never have a point
in the closure of the singleton set
without the point
being in the closure of
. Adding
to this condition gives us
. A
space is one in which any two distinct points are not only topologically distinguishable, but separated. In particular, we can see that the singleton set
is closed, since its closure can’t contain any other points than
itself.
A space is if any two topologically distinguishable points are separated by neighborhoods. If this also holds for any pair of distinct points we say that the space is
, or “Hausdorff”. This is where most topologists start to feel comfortable, though the topologies that arise in algebraic geometry are usually non-Hausdorff. To a certain extent (well, to me at least) Hausdorff spaces feel a lot more topologically natural and intuitive than non-Hausdorff spaces, and you almost have to try to construct pathological spaces to violate this property. Back in graduate school, some of us adapted the term to apply more generally, as in “That guy Steve is highly non-Hausdorff.”
One interesting and useful property of Hausdorff spaces is that the image of the diagonal map defined by
is closed. To see this, notice that it means the complement of the image is open. That is, if
is a pair of points of
with
then we can find an open neighborhood containing the point
consisting only of pairs
with
. In fact, we have a base for the product topology on
consisting of products two open sets in
. That is, we can pick our open neighborhood of
to be the set of all pairs
with
and
, where
is an open subset of
containing
and
is an open subset containing
. To say that this product doesn’t touch the diagonal means that
, which is just what it means for
and
to be separated by neighborhoods!
We can strengthen this by asking that any two distinct points are separated by closed neighborhoods. If this holds we say the space is . There’s no standard name for the weaker version discussing topologically distinguishable points. Stronger still is saying that a space is “completely Hausdorff” or completely
, which asks that any two distinct points be separated by a function.
A space is “regular” if given a point
and a closed subset
with
we can separate
and
by neighborhoods. This is a bit stronger than being Hausdorff, where we only asked that this hold for two singletons. For regular spaces, we allow one of the two sets we’re separating to be any closed set. If we add on the
condition we’re above
, and so singletons are just special closed sets anyhow, but we’re strictly stronger than regularity now. We call this condition
.
As for Hausdorff, we say that a space is completely regular if we can actually separate and
by a function. If we take a completely regular space and add
, we say it’s
, or “completely regular Hausdorff”, or “Tychonoff”.
We say a space is “normal” if any two disjoint closed subsets are separated by neighborhoods. In fact, a theorem known as Urysohn’s Lemma tells us that we get for free that they’re separated by a function as well. If we add in (not
this time) we say that it is “normal Hausdorff”, or
.
A space is “completely normal” if any two separated sets are separated by neighborhoods. Adding in we say that the space is “completely normal Hausdorff”, or
.
Finally, a space is “perfectly normal” if any two disjoint closed sets are precisely separated by a function. Adding makes the space “perfectly normal Hausdorff”, or
.
The Wikipedia entry here is rather informative, and has a great schematic showing which of the axioms imply which others. Most of these axioms I won’t be using, but it’s good to have them out here in case I need them.