Is there exists a locally compact topological space $X$, for every compact subset $Ksubseteq X$, we have $K$ is not closed?

Or slightly weaker, is there a locally compact topological space $X$, such that there exists $xin X$, with the property that all compact neighborhood of $x$ is not closed?

If $X$ is a locally compact topological group, the answer to the above question is false, because all neighborhood contains a closed neighborhood, so I want to know more general situation, thanks for any helpful answers, any comment is welcome!