Some observations on compact indestructible spaces

Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelöf T2 indestructible space has the finite derived set property and a compact T2 indestructible space is pseudoradial. Finally, we observe that under CH a compact weakly Whyburn space of countable tightness is indestructible.