The Arens’ space is a canonical example of a sequential space that is not a Frechet space. It also has a subspace that is not sequential (thus the notion of being a sequential is not hereditary). We show that any space that is sequential but not Frechet contains a copy of the Arens’ space. For previous discussion on sequential spaces and Frechet spaces, see the links at the end of this post. Also see [1] and [2].
Let be the set of all nonnegative integers. Let
be the set of all positive integers. In one formulation, the Arens’ space is the set
with the open neighborhoods defined by:
- The points in
are isolated;
- The neighborhoods at each
are of the form
for some
;
- The neighborhoods at
are obtained by removing from
finitely many
and by removing finitely many isolated points in each of the remaining
.
Another formulation is that of a quotient space. For each , let
be a convergent sequence such that
is the limit. Let
be a topological sum of the convergent sequences
. We then identify
for each
. The Arens’ space is the resulting quotient space and let
denote this space (in the literature
is used). Note that the Arens’ space has been previously defined in this blog (see An example of a quotient space, II). Note that the quotient space
is topologically identical to
. In the remainder of this note, we work with
in discussing the Arens’ space.
The Arens’ space is sequential since it is a quotient space of a first countable space. The subspace is not sequential, proving that the Arens’ space is not a Frechet space.
We now show that any sequential space that is not Frechet contains a copy of the Arens’ space. We have the following theorem.
Theorem
Let be a sequential space. Then
is Frechet if and only
does not contain a copy of the Arens’ space.
Proof
This direction is clear since the Frechet property is hereditary.
For any
, let
be the set of limits of sequences of points of
. Suppose
is not Frechet. Then for some
, there exists
such that
. Since
is non-closed in
and since
is sequential, there is a sequence
of points of
converging to
. We can assume that
for all but finitely many
(otherwise
). Thus without loss of generality, assume
for all
.
For each , there is a sequence
of points of
converging to
. It is OK to assume that all
are distinct and all
are distinct across the two indexes. Let
where
and
. Then
is a homeomorphic copy of the Arens’ space.
Remark
The above theorem is not valid outside of sequential spaces. Let be a countable space with only one non-isolated point where
is not sequential (for example, the subspace
of the Arens’ space). Clearly
contains no copy of the Arens’ space. Yet
is not Frechet (it is not even sequential).
Previous posts on sequential spaces and Frechet spaces:
Sequential spaces, I
Sequential spaces, II
Sequential spaces, III
Sequential spaces, IV
Sequential spaces, V
k-spaces, I
k-spaces, II
Reference
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.