In this paper we further investigate the results given in [10], [11], [12]. In Section 2 we consider $\Sigma C$-filters (ultrafilters). Let $X$ be a $\sigma$-compact, dense subspace of a locally compact space $Y$. The space $Y$ is compact if and only if every $\Sigma C$-ultrahlter on $X$ converges to some point in $Y$. In Section 3 we consider $P$-sets, $\Sigma C$-filters (ultrafilters) and $HCC$ property. A locally compact space $X$ is $HCC$ if and only if every $\Sigma C$ ultrafilter on $X$ converges. In section 3 we also consider $HCC$ extensions of locally compact spaces.