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Open subspace of a compact space topological space

  1. Mar 8, 2009 #1
    It is a fact that if X is a compact topoloical space then a closed subspace of X is compact.
    Is an open subspace G of X also compact?
    please consider the following and note if i am wrong;

    proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the union of all sets in this class is G. but X is compact and each H_i is the intersection of G with an open subset P_i of X for corresponding i. The result follows from the fact {p_i} has a finite subclass which contains X.
    hence every open subspace of a compact space is compact.

    pls, am i right?
  2. jcsd
  3. Mar 8, 2009 #2


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    Staff Emeritus
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    Gold Member

    Your proof is wrong, as you can have an open cover of G in the topology of X that doesn't cover X, and then when you reduce it to the induced topology on G it doesn't need to have a finite subcover.

    Conclusion: All you did is prove that an open cover of a compact space is also an open cover of its subsets (not a very impressive result :p )

    As a quick counterexample, see (0,1) contained in [0,1]
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