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 a topological embedding.

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

Then either $f$ or $(Y, \tau_2)$ may be called a compactification of $(X,\tau_1)$.

The latter case can be confusing under certain circumstances. Its use should usually be limited to one of the following situations:
 * $X \cap Y = \varnothing$
 * $(X,\tau_1)$ is a subspace of $(Y,\tau_2)$ and $f$ is the inclusion mapping.

Also defined as
Many writers require the space $Y$ to be a Hausdorff space.

Some writers do not require density.

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