We call a topological space $\kappa$-compact if every subset of size $\kappa$ has a complete accumulation point in it. Let $\Phi (\mu ,\kappa ,\lambda )$ denote the following statement: $\mu <\kappa <\lambda =cf\left(\lambda \right)$ and there is $\{S_{\xi }:\xi <\lambda \}\subset {\left[\kappa \right]}^{\mu }$ such that $\left|\right\{\xi :|S_{\xi }\cap A|=\mu \left\}\right|<\lambda$ whenever $A\in {\left[\kappa \right]}^{<\kappa }$. We show that if $\Phi (\mu ,\kappa ,\lambda )$ holds and the space $X$ is both $\mu$-compact and $\lambda$-compact then $X$ is $\kappa$-compact as well. Moreover, from PCF theory we deduce $\Phi (cf\left(\kappa \right),\kappa ,{\kappa }^{+})$ for every singular cardinal $\kappa$. As a corollary we get that a linearly Lindelöf and ${\aleph }_{\omega }$-compact space is uncountably compact, that is $\kappa$-compact for all uncountable cardinals $\kappa$.

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z)\ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of ${\ell }_{\infty }$.