We introduce a new concept - $\tau$-extendability of topological properties, and prove that certain compactness-type properties are $\tau$-extendable for uncountable $\tau$. Answering D. B. Shakhmatov's question we present for any cardinal $\lambda$ an example of a pseudocompact space $X$ with $G_\delta$-diagonal and $|X|\geq\lambda$. Under CH this space $X$ can be made 2-pseudocompact which is stronger than being pseudo-compact. A ZFC example of a 2-pseudocompact space which has no dense relatively countably compact subspace is also given.