the continuous image of a compact space is compact
To see this remember, since the continuity of implies that each is open, all we need to prove is that . Consider , we know that since is a covering of that there exists such that but then by construction and is indeed an open covering of .
To see that is a covering of consider . By the surjectivity of there must exist (at least) one such that and since is a finite covering of , there exists such that . But then since , we must have that and is indeed a finite open covering of .
|Title||the continuous image of a compact space is compact|
|Date of creation||2013-03-22 15:52:48|
|Last modified on||2013-03-22 15:52:48|
|Last modified by||cvalente (11260)|