if F is aclosed subset of acompact set K in R then F is compact?

2 Answers

  • holdm
    Lv 7
    1 decade ago
    Favorite Answer

    Let {Oi} be an open cover of F. Then {Oi + -F} is an open over of K, since F closed implies -F open. Therefore there exists a finite subcover S of {Oi+ -F}. Remove -F and this is a finite subcover of F. For, let x be in F. Then x is in some open set in S. But x is not in -F. Therefore x is in some Oi in S. and the Oi's of S form a finite subcover of F.

    Therefore F is compact.

  • squeek
    Lv 4
    1 decade ago

    Since K is compact in R, it is closed and bounded. Therefore, F must also be bounded.

    Since F is a closed and bounded subset of R, then F is compact.

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