Definition:Compactification

Definition

Let $\struct {X, \tau_1}$ be a topological space.

Let $\struct {Y, \tau_2}$ be a compact space.

Let $f: X \to Y$ be a topological embedding.

Let $\Img f$ be everywhere dense in $Y$.

Then either $f$ or $\struct {Y, \tau_2}$ may be called a compactification of $\struct {X, \tau_1}$.

The latter case can be confusing under certain circumstances.

Its use should usually be limited to one of the following situations:

$(1): \quad X \cap Y = \O$

$(2): \quad \struct {X, \tau_1}$ is a subspace of $\struct {Y, \tau_2}$ and $f$ is the inclusion mapping.

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.