# One-point compactification

1. Apr 28, 2012

### Flying_Goat

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

Suppose (X,τx) is a topological space. Define Y=X$\cup${p} and a new topology τY such that U$\subseteq$Y 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,τY) is compact, it seems to require X-U in (2) to be compact under τY and not τX. That is if p$\in$ U, then any open covering {Vi}$\subseteq$τY of Uc=X-U, has a finite subcover. But we don't even know what our open sets {Vi} are in the first place.

Any help would be appreciated.

2. Apr 28, 2012

### Office_Shredder

Staff Emeritus
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

3. Apr 28, 2012

### Flying_Goat

Ah, I see. Thanks very much.