We construct an ultrafilter $u \in \omega\setminus\omega^*$ such that the subspace $\omega^*\setminus\{u\}$ is not absolutely countably compact, and we show that under the continuum hypothesis, for every $u \in \omega\setminus\omega^*$ the subspace $\omega^*\setminus\{u\}$ is not absolutely countably compact.