Universal definition of the indiscrete topology
Exercise 182 (Chapter 29, “The category of topological spaces”):
Give a universal definition which leads to the introduction of the indiscrete topology on a set.
On a set , the indiscrete topology is the unique topology such that for any set
, any mapping
is continuous.
This isn’t really a universal definition. We have seen that the discrete topology can be defined as the unique topology
that makes
a free topological space on the set
. We will give the indiscrete topology, which is at the other end of the spectrum of topologies, a similar definition.
For any category , we can define a “reversed arrows” category
, which has the same objects as
and the same morphisms, except that the sources and targets of the morphisms are reversed. Composition too is, obviously, reversed; the identities are the same. It is easily checked that this does lead to a proper category, and also that the arrows-reversed category of an arrows-reversed category leads back to the original category.
If we have categories and
and a covariant functor
from
to
, then we can consider the three modified functors
from
to
,
from
to
and
from
to
. These functors will map both objects and morphisms exactly as the original
does. Those with only one reversal are contravariant; the one with a reversal in both categories is covariant, and is the one that will interest us here.
We consider the classical category of sets ; in its arrows-reversed version,
, the morphisms from set
to set
are the mappings from
to
. Similarly, we have the category of topological spaces
and its arrows-reversed version
in which morphisms from topological space
to topological space
are the continuous mappings from
to
.
For a given set , object of
, we can ask if we can find a free object
in category
following the forgetful functor
from
to
. This mens finding a topological space
and a
-morphism
such that for any topological space
and any
-morphism
, there exists a unique
-morphism
such that (as
-morphisms)
.
Factoring in the arrows-reversals, this means that we wish to find a topological space and a mapping
such that for any topological space
and any mapping
, there exists a unique continuous mapping
such that (as mappings)
.
We propose that the topological space with
the indiscrete topology on
, together with the mapping
, is a free object on
following the functor
.
Proof:
Let be a topological space and
a mapping
. If
is to be a continuous mapping
such that
, it must be a mapping
such that
; that is, we must have
. This
is indeed continuous, since
is the indiscrete topology. Hence it is the unique continuous mapping
such that
.
Thus we have shown that with the indiscrete topology, together with the identity mapping on
, is a free topological space following the arrows-reverse categorical definition of free objects.
Free objects are unique up to an isomorphism. However, we wish to show more specifically that given the set , the indiscrete topology
on
is the unique topology
such that
be a free object on
following the above definition. We could use the “unique up to an isomorphism” card, but more simply: If
is to be a free object on
following the above definition, in particular, taking
and
, we need there to exist a continuous
, that is,
such that
, that is such that
. In other words, we need
to be continuous from
to
, which implies
, that is, that
must be coarser than
; which is possible only if
.
Hence the indiscrete topology on is the unique topology
such that
be a free object in category
following the functor
from category
to category
.