Abstract:We prove some closed mapping theorems on $k$-spaces with point-countable $k$-networks. One of them generalizes La\v snev's theorem. We also construct an example of a Hausdorff space $Ur$ with a countable base that admits a closed map onto metric space which is not compact-covering. Another our result says that a $k$-space $X$ with a point-countable $k$-network admitting a closed surjection which is not compact-covering contains a closed copy of $Ur$.
Keywords: $k$-space, $k$-network, closed map, compact-covering map
AMS Subject Classification: 54B10, 54A20