gn.general topology – a locally compact space which has no closed subset

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!