Around tight points

We present three examples of countable spaces with a single nonisolated point. The first gives, assuming CH, a Fréchet-Urysohn tight point which does not have countable absolute tightness; the second, constructed with the help of ◊, gives a tight non-weakly Fréchet-Urysohn point without countable absolute tightness; the third gives a weakly Fréchet-Urysohn point which is not Fréchet-Urysohn, has countable fan-tightness but is not tight.