Tag Archives: Arens' Space

Defining Arens’ space using diagrams

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One way to define the Arens’ space is a 2-step approach, which is the quotient space approach. The first step is to identify an Euclidean plane consisting of convergent sequences (usually conveniently situated in the two-dimensional plane). The second step is to collapse certain points to make it a quotient space. Another way is to define the space directly, usually using an appropriate subset of the plane (of course, the resulting space is not an Euclidean space). We demonstrate both approaches using diagrams. In the first approach, we use two diagrams, the first one showing what the Euclidean space should look like, the second showing the resulting Arens’ space after certain points are identified. In the second approach, only one diagram is used (the standalone approach). The two-step approach is actually more informative since the quotient space of a separable metric space is a sequential space.

The following diagrams define the spaces without identifying specific points or locations in the Euclidean plane. The diagrams only indicate how the points relate to one another. For a definition of Arens’ space using the quotient space approach using specific points in the plane, see here. For a definition without connection to quotient space, see here. The red diagram and the blue diagram are for the quotient space approach (two-step). The pink diagram is the standalone approach.

The Arens’ space as discussed here is related to the Arens-Fort space, example 26 in Counterexamples in Topology [2].

The Red Diagram – The Euclidean Space

      \displaystyle \begin{aligned} & \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & S_5 & \text{ } & S_4 & \text{ } & S_3 & \text{ } & S_2 & \text{ } & S_1 \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } &\bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot &\text{ } &\cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot &\text{ } & \cdot & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & q_5 &\text{ } & q_4 & \text{ } & q_3 &\text{ } & q_2 & \text{ } & q_1 \\& \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\& \bullet & \leftarrow & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet \\& p & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & p_5 &\text{ } & p_4 & \text{ } & p_3 &\text{ } & p_2 & \text{ } & p_1 \end{aligned}

In all 3 diagrams, the thick bullets represent points in the space and the tiny dots represent “dot dot dot”, indicating that the points or the sequences are continuing countably infinitely. The arrows in the diagrams point to the direction to which the points are converging.

The points in the red diagram form a subspace of the Euclidean plane. There are convergent sequences S_n going downward and going across from right to left. The point q_n is the limit of the sequence S_n. The sequence of points q_n converges to a point, which is ignored and not shown in the diagram. The points p_n are situated below the points q_n and converge to the point p. In this Euclidean space, the points in the sequences S_n are isolated points. An open set of the point q_n consists of q_n and all but finitely many points in the sequence S_n. Each point p_n is isolated. An open set of the point p consists of p and all but finitely many p_n.

The Blue Diagram – The Arens’ Space as a Quotient Space

      \displaystyle \begin{aligned} & \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & S_5 & \text{ } & S_4 & \text{ } & S_3 & \text{ } & S_2 & \text{ } & S_1 \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } &\bullet & \text{ } & \bullet & \text{ } & \bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet & \text{ } &\bullet & \text{ } & \bullet \\& \text{ } & \text{ } & \cdot &\text{ } &\cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot &\text{ } & \cdot & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow & \text{ } & \downarrow \\& \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet \\& p & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & p_5 &\text{ } & p_4 & \text{ } & p_3 &\text{ } & p_2 & \text{ } & p_1 \end{aligned}

The blue diagram is established from the red diagram. The blue diagram is obtained by identifying the points q_n and p_n in the red diagram as one point called p_n. The resulting quotient space is the Arens’ space. Let S be the set of all points in the sequences S_n. Let P be the set of all points p_n. The Arens’ space is X=S \cup P \cup \{ p \} with the quotient topology. With the quotient topology, an open set containing the point p is obtained by removing finitely many vertical columns (a column is S_n plus the point p_n) from X and by removing finitely many points in the remaining columns. An open neighborhood of the point p_n consists of p_n and all but finitely many points in the sequence S_n. Points in the sequences S_n continue to be isolated points.

The Arens’ space is a sequential space since it is the quotient image of a separable metric space.

The Pink Diagram – The Arens’ Space as a Standalone Space

        \displaystyle \begin{aligned} & \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ }& \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot &\text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & S_1 & \text{ } & S_2 & \text{ } & S_3 & \text{ } & S_4 & \text{ } & S_5 & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } &\text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\& \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet &\text{ } & \bullet & \text{ } & \bullet & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \\& p & \text{ } & p_1 &\text{ } & p_2 & \text{ } & p_3 &\text{ } & p_4 & \text{ } & p_5 & \text{ } & \cdot & \text{ } & \cdot & \text{ } & \cdot & \text{ } \end{aligned}

In the pink diagram, the bottom rows are the limit points. The point p_n is the sequential limit of the sequence S_n. The sequence S_n is displayed vertically. The convergence of the sequence S_n is not exhibited in the diagram and follows from how the open sets are defined.

Let S be the set of all points in the sequences S_n. Let P be the set of all points p_n. The Arens’ space is X=S \cup P \cup \{ p \} with the topology defined as follows. Each point in S is an isolated point. An open neighborhood of p_n \in P consists of the point p_n and all but finitely many points in S_n. An open neighbood containing the point p is obtained by removing finitely many vertical columns (a column is S_n plus the point p_n) from X and by removing finitely many points in the remaining columns.

Remarks

The blue diagram and the pink diagram are both representations of the Arens’ space. The space consists of countably many convergent sequences and their limits points plus one additional limit point called p. Both diagrams present the essential ideas without being tied to specific points or sequences in the Euclidean plane. Perhaps the diagrams will make it easier to think about the Arens’ space and remember the definition.

As mentioned earlier, the Arens’ space is a sequential space since it is the quotient space of a metric space (see the theorem here). Recall from above that the Arens’ space is X=S \cup P \cup \{ p \}. Clearly, p \in \overline{S}. Note that no sequence of points in S can converge to the point p. Thus, the Arens’ space is an example of a sequential space that is not a Frechet space. In fact, in order to know whether a sequential space W is Frechet, all we need to do is to determine if W contains a copy of the Arens’ space (see the theorem here). Thus, any space that is sequential but not Frechet contains a copy of the Arens’ space. In this case, Frechetness is characterized the absence of an Arens’ subspace. The Arens’ space is a canonical quotient space that appears in the characterization of other properties. See [1] for an example.

The property of being a sequential space is not hereditary. Consider the subspace of the Arens’ space Y=S \cup \{ p \}. As observed in the preceding paragraph, no sequence of points in S can converge to p. Thus, the set S is a sequentially closed set but not closed in Y. This means that Y is not a sequential space. Thus, the Arens’ space is a sequential that is not hereditarily sequential. In fact, a space is a Frechet space if and only if it is a hereditarily sequential space (see Theorem 1 here).

The subspace Y=S \cup \{ p \} discussed in the preceding paragraph is the Arens-Fort space, which is the example 26 in Steen and Seebach [2].

Reference

  1. Lin, S., A note on the Arens’ space and sequential fan, Topology Appl, 81, 185-196, 1997.
  2. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.

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