A note about the Arens’ space

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 \omega be the set of all nonnegative integers. Let \mathbb{N} be the set of all positive integers. In one formulation, the Arens’ space is the set X=\left\{\infty\right\} \cup \mathbb{N} \cup (\mathbb{N} \times \mathbb{N}) with the open neighborhoods defined by:

  • The points in \mathbb{N} \times \mathbb{N} are isolated;
  • The neighborhoods at each n \in \mathbb{N} are of the form B_{n,m}=\left\{n\right\} \cup \left\{(n,j) \in \mathbb{N} \times \mathbb{N}:j \ge m\right\} for some m \in \mathbb{N};
  • The neighborhoods at \infty are obtained by removing from X finitely many B_{n,1} and by removing finitely many isolated points in each of the remaining B_{n,1}.

Another formulation is that of a quotient space. For each n \in \omega, let K_n=\left\{x_{n,j}:j \in \mathbb{N}\right\} \cup \left\{y_n\right\} be a convergent sequence such that y_n is the limit. Let G be a topological sum of the convergent sequences K_n. We then identify \left\{x_{0,j},y_j\right\} for each j \in \mathbb{N}. The Arens’ space is the resulting quotient space and let Y denote this space (in the literature S_2 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 Y is topologically identical to X. In the remainder of this note, we work with X in discussing the Arens’ space.

The Arens’ space is sequential since it is a quotient space of a first countable space. The subspace \left\{\infty\right\} \cup (\mathbb{N} \times \mathbb{N}) 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 W be a sequential space. Then W is Frechet if and only W does not contain a copy of the Arens’ space.

Proof
\Longrightarrow This direction is clear since the Frechet property is hereditary.

\Longleftarrow For any T \subset W, let T^s be the set of limits of sequences of points of T. Suppose W is not Frechet. Then for some A \subset W, there exists x \in \overline{A} such that x \notin A^s. Since A^s is non-closed in W and since W is sequential, there is a sequence w_n of points of A^s converging to z_0 \notin A^s. We can assume that w_n \notin A for all but finitely many n (otherwise z_0 \in A^s). Thus without loss of generality, assume w_n \notin A for all n.

For each n \in \mathbb{N}, there is a sequence z_{n,j} of points of A converging to w_n. It is OK to assume that all w_n are distinct and all z_{n,j} are distinct across the two indexes. Let W_0=\left\{z_0\right\} \cup W_1 \cup W_2 where W_1=\left\{w_n: n \in \mathbb{N}\right\} and W_2=\left\{z_{n,j}:n,j \in \mathbb{N}\right\}. Then W_0 is a homeomorphic copy of the Arens’ space. \blacksquare

Remark
The above theorem is not valid outside of sequential spaces. Let Z be a countable space with only one non-isolated point where Z is not sequential (for example, the subspace Z=\left\{\infty\right\} \cup (\mathbb{N} \times \mathbb{N}) of the Arens’ space). Clearly Z contains no copy of the Arens’ space. Yet Z 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

  1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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