An observation about sequential spaces

This post is about an observation about sequential spaces. In a sequential space, non-trivial convergent sequences abound. Thus in the extreme case of there being no trivial convergent convergent sequences, the space in question must not be sequential. Specifically we observe that if X is a Hausdorff sequential space and if p \in X is a non-isolated point (i.e. the singleton set \left\{p\right\} is not open), there is a convergent sequence p_n of points of X-\left\{p\right\} such that p_n \mapsto p. Thus it is necessary condition that in a sequential space, there exist non-trivial convergent sequences at every non-isolated point. We present examples showing that this condition is not a sufficient condition for a space being sequential. As the following examples show, the property that there are non-trivial convergent sequences at every non-isolated is a rather weak property.

The first example is from the problem section of Mathematical Monthly in 1970 (see [2]). Let \mathbb{R} be the real line and let \mathbb{P} be the set of all irrational numbers. Let \mathbb{Q}=\mathbb{R}-\mathbb{P}. Let X=\mathbb{R} and define a new topology on X by calling a subset U \subset X open if and only if U=W-H where W is a usual open subset of the real line and H is a subset of \mathbb{P} that is at most countable. This new topology on the real line is finer than the Euclidean topology. Thus X is a Hausdorff space. Every point of X is a non-isolated point and is the the sequential limit of a sequence of rational numbers, satisfying the condition that every non-isolated point is the sequential limit of a non-trivial convergent sequence.

In the topology for X, every countably infinite subset of the set \mathbb{P} is closed in X. Thus no sequence of points of \mathbb{P} can converge to a point not in \mathbb{P}. Therefore \mathbb{P} is sequentially closed and non-closed in X, making X not a sequential space.

Not only that every countably infinite subset of \mathbb{P} is closed in X, every countably infinite subset of \mathbb{P} is relatively discrete. Then it follows that for every compact K \subset X, K \cap \mathbb{P} is finite (and is thus closed in K). Thus X is also not a k-space.

Another example is that of a product space. Any uncountable product where each factor has at least two points is not sequential. This follows from the fact that 2^{\omega_1} is not sequential (see Sequential spaces, IV). Furthermore, in any product space with infinitely many factors each of which has at least two points, every point is the sequential limit of a non-trivial convergent sequence. Thus any product space with uncountably many factors, each of which has at least two points, is another example of a non-sequential space where there are non-trivial convergent sequences at every point.

Previous posts on sequential spaces and k-spaces:
Sequential spaces, I
Sequential spaces, II
Sequential spaces, III
Sequential spaces, IV
Sequential spaces, V
k-spaces, I
k-spaces, II
A note about the Arens’ space

Reference

  1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Henkel, D. Solution to Monthly Problem 5698, American Mathematical Monthly 77, p. 896, 1970
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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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.

k-spaces, II

A space X is a k-space if for each A \subset X, A is closed in X if and only if K \cap A is closed in K for all compact K \subset X. A space X is a sequential space if for each A \subset X, A is closed in X if and only if A is a sequentially closed set in X. A set A \subset X is sequentially closed in the space X if whenever we have x_n \in A and the sequence x_n converges to x \in X, we have x \in A. A set A \subset X is sequentially open in X if X-A is sequentially closed in X. In both of these definitions, we can replace “closed” with “open” and the “only if” part of the definition always hold. Thus in working with these definitions, we only need to be concerned with the “if” part. Every sequential space is a k-space. The converse does not hold. In this short note, we show that the converse holds if every point in the space is a G_\delta-set. This is a basic fact about k-spaces. For other basic facts on k-spaces and sequential spaces, see the following:

Sequential spaces, I
Sequential spaces, II
Sequential spaces, III
Sequential spaces, IV
Sequential spaces, V
k-spaces, I

In a given space X, W \subset X is a G_\delta-set in X if W=\bigcap \limits_{i=1}^\infty U_i where each U_i is open in X, i.e. W is the intersection of countably many open sets. A point x \in X is a G_\delta-set in X if the singleton \left\{x\right\} is the intersection of countably many open subsets of X. It is a well known fact in general topology that in a compact Hausdorff space X, if x \in X is a G_\delta-set in X, then there is a countable local base at x. It follows that if every point of a compact Hausdorff space X is a G_\delta-set in X, then X is first countable (see The cardinality of compact first countable spaces, II).

Theorem
Let X be a space in which every point is a G_\delta-set in X. Then if X is a k-space then X is a sequential space.

Proof. Suppose A \subset X is not closed in X. We show that A is not sequentially closed in X, i.e. there is a sequence x_n \in A such that x_n \mapsto x \in X and x \notin A.

Since X is a k-space and A is not closed, there is a compact K \subset X such that K \cap A is not closed in K. Every point of K is a G_\delta-set in X and thus a G_\delta-set in K. It follows that K is first countable.

Let x \in \overline{K \cap A} such that x \notin A (the closure is taken in K). Since K is first countable, there is a sequence x_n \in K \cap A such that x_n \mapsto x. This means A is not sequentially closed in X. \blacksquare