- #1
Flying_Goat
- 16
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Hi guys, I am confused about the definition of compactification of a topological space.
Suppose (X,τx) is a topological space. Define Y=X[itex]\cup[/itex]{p} and a new topology τY such that U[itex]\subseteq[/itex]Y is open if
(1) p [itex]\notin[/itex] U and U[itex]\in[/itex] [itex]\tau[/itex]X or
(2) p [itex]\in[/itex] 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[itex]\in[/itex] U, then any open covering {Vi}[itex]\subseteq[/itex]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.
Suppose (X,τx) is a topological space. Define Y=X[itex]\cup[/itex]{p} and a new topology τY such that U[itex]\subseteq[/itex]Y is open if
(1) p [itex]\notin[/itex] U and U[itex]\in[/itex] [itex]\tau[/itex]X or
(2) p [itex]\in[/itex] 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[itex]\in[/itex] U, then any open covering {Vi}[itex]\subseteq[/itex]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.