Arens square


In mathematics, the Arens square is a topological space.

Definition

Arens square is the topological space where
The topology is defined from the following basis. Every point of is given the local basis of relatively open sets inherited from the Euclidean topology on. The remaining points of are given the local bases

Properties

The space satisfies that:
  1. is T, since neither points of, nor, nor can have the same second coordinate as a point of the form, for.
  2. is not T or T, since for there is no open set such that since must include a point whose first coordinate is, but no such point exists in for any.
  3. is not Urysohn, since the existence of a continuous function such that and implies that the inverse images of the open sets and of with the Euclidean topology, would have to be open. Hence, those inverse images would have to contain and for some. Then if, it would occur that is not in. Assuming that, then there exists an open interval such that. But then the inverse images of and under would be disjoint closed sets containing open sets which contain and, respectively. Since, these closed sets containing and for some cannot be disjoint. Similar contradiction arises when assuming.
  4. is semiregular, since the basis of neighbourhood that defined the topology consists of regular open sets.
  5. is second countable, since is countable and each point has a countable local basis. On the other hand is neither weakly countably compact, nor locally compact.
  6. is totally disconnected but not totally separated, since each of its connected components, and its quasi-components are all single points, except for the set which is a two-point quasi-component.
  7. is not scattered, since each basis set is dense-in-itself.
  8. is not zero-dimensional, since doesn't have a local basis consisting of open and closed sets. This is because for small enough, the points would be limit points but not interior points of each basis set.