If $X$ a topological space one says that $X$ is *universally closed* if for every Hausdorff space $Y$ and every (continuous) map $f:X\rightarrow Y$, the image of $X$ is a closed subset of $Y$.


It is clear that every compact space is universally closed, but are there non compact universally closed spaces?



Topological spaces whose continuous image is always closed
If $Z$ is not compact, and $X=\{p\}\cup Z$ is the space whose nonempty open sets are of the form $\{p\}\cup V$ with $V$ open in $Z$, then $X$ is not compact, but every continuous function from $X$ to a Hausdorff space is constant.


