Definition:Compactification

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Definition

Let $\left({X, \tau_1}\right)$ be a topological space.

Let $\left({Y, \tau_2}\right)$ be a compact space.

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

Let $f \left({X}\right)$ be everywhere dense in $Y$.


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


Warning

The latter case can be confusing under certain circumstances.

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

$(1): \quad X \cap Y = \varnothing$
$(2): \quad \left({X, \tau_1}\right)$ is a subspace of $\left({Y, \tau_2}\right)$ 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.