Hi guys, I am confused about the definition of compactification of a topological space.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose (X,τ_{x}) is a topological space. Define Y=X[itex]\cup[/itex]{p} and a new topology τ_{Y}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,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[itex]\in[/itex] U, then any open covering {V_{i}}[itex]\subseteq[/itex]τ_{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.

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# One-point compactification

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