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  and .
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.
Let be a sequential space. Then is Frechet if and only does not contain a copy of the Arens’ space.
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.
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).
- Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.