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Suppose $X$ has two topologies $(X,T),(X,T')$ and both are compact and hausdorffs. Suppose $T \subset T'$. Prove that $T=T'$

Because of the compactness of both topologies and the fact that one is an open subset of the other, I can clearly see that we can use the same finite open coverings to cover these topologies. But how do i use the hausdorff definition in this question?

Kees Til

Kees Til
  • 1,978

1 Answers1

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Show that the identity $(X, T') \to (X, T)$ is continuous. Note that it's a bijection from a compact space to a Hausdorff space.

Ayman Hourieh
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  • Don't I get problems with elements which are in the topology of T' and not in the topology of T, or do this topologies only determine if elements of X are closed/open?

    I see that i can make a homeomorphism now, which helps a lot :)

     – Kees Til May 29 '14 at 11:43
  • You want to show that $f^{-1}(V)$ is open in $T'$ if $V$ is open in $T$. Note that $T \subset T'$. You only need this direction in order to show continuity. – Ayman Hourieh May 29 '14 at 11:46
  • True that! i just realized what a stupid question that was, thank you for your time :) – Kees Til May 29 '14 at 11:47
  • You're welcome. – Ayman Hourieh May 29 '14 at 11:50