Compactification

Definition

Consider a Hausdorff Space $X$ and a Compact Hausdorff Space $Y$. The $Y$ is a compactification of $X$ if $X$ is contained in $Y$ as a subspace such that $\bar{X}=Y$ (Dense Set). Or equivalently $(Y,e)$ is a compactification of $X$ if there exists an embedding $e:X\hookrightarrow Y$ whose image $e(X)$ is a dense subspace of $Y$. $\exists e:X\hookrightarrow Y\text{s.t.}\overline{e(X)}=Y$