What is the Definition of Compactification for a Topological Space?

[Flying_Goat]

Hi guys, I am confused about the definition of compactification of a topological space.

Suppose (X,τ_x) is a topological space. Define Y=X\cup{p} and a new topology τ_Y such that U\subseteqY is open if
(1) p \notin U and U\in \tau_X or
(2) p \in U and X-U is a compact closed subset of X.

To prove that (Y,τ_Y) is compact, it seems to require X-U in (2) to be compact under τ_Y and not τ_X. That is if p\in U, then any open covering {V_i}\subseteqτ_Y of U^c=X-U, has a finite subcover. But we don't even know what our open sets {V_i} are in the first place.

Any help would be appreciated.

 

[Office_Shredder]

The induced topology of X as a subspace of Y is the same as the original topology of X (this is not obvious, but not hard to prove, just some definition chasing), so if we have
A\subset X \subset Y

A is compact in the topology of Y if and only if it is compact in the topology of X

 

[Flying_Goat]

Ah, I see. Thanks very much.