Definition:Compactification

Definition
Let $(X,\tau_1)$ be a topological space.

Let $(Y, \tau_2)$ be a compact space.

Suppose that $f:X \to Y$ is an embedding.

Suppose that $f(X)$ is dense in $Y$.

Then $f$ is a compactification of $X$.

Alternative Definitions
Some writers do not require density.

Some writers describe constructs as "compactifications" though those constructs may not be compact in all circumstances.