We just had the definition of compactification in our lecture, which says that Y must be compact and X must be a dense and open subspace of X. However in his Notes he gave the Definition so that X must only be a dense, open subset. My question now is if this condition is necessary for compactification or if it can be put down.
I am especially interested if compact Sets wouldnt necessarily remain compact if X only needs to be a subset.
On
No, if $Y$ is a compactification of $X$, then $X$ needs to be a subspace of $Y$. That is, the topology that $X$ had to begin with must coincide with its topology as a subspace of $Y$.
On the other hand, the definition of "compactification" that I'm accustomed to does not require $X$ to be open in $Y$. I think that will be the case in general only when $X$ is locally compact.
The term "subspace" in the context of topology means, essentially, a subset when considered with the induced topology.
In particular, every subspace is a subset of the space, and every subset is a subspace of the space.