A counterexample in the theory of $D$-spaces

TL;DR

Under the assumption of iamondsuit, the paper constructs a hereditarily Lindelf6f space of size 1 that is not a D-space, with all finite powers also Lindelf6f, providing a counterexample in the theory of D-spaces.

Contribution

The paper presents the first known example of a hereditarily Lindelf6f space of size 1 that is not a D-space, assuming iamondsuit.

Findings

Constructed a hereditarily Lindelf6f space of size 1 that is not a D-space

All finite powers of the space are Lindelf6f

The example relies on the iamondsuit assumption

Abstract

Assuming $\diamondsuit$, we construct a $T_2$ example of a hereditarily Lindel\"of space of size $\omega_1$ which is not a $D$-space. The example has the property that all finite powers are also Lindel\"of.