Space S000047

Countable sum of Sierpinski spaces

Also known as: Hjalmar Ekdal topology

Let XX be the set of positive integers and define a topology by declaring UXU \subseteq X to be open if for every odd pUp \in U, p+1Up+1 \in U. Equivalently, AXA \subseteq X is closed iff for every even pAp \in A, p1Ap-1 \in A.

This space is a topological sum of countably-many copies of the Sierpinski space.

Defined as counterexample #55 ("Hjalmar Ekdal Topology") in DOI 10.1007/978-1-4612-6290-9.

A piece of trivia: the origin of the name Hjalmar Ekdal is explained here.