If a topological space X is compact Hausdorff then it is normal but is X completely normal too. I can't think of any reason why this should be so but then again I can't come up with a counterexample. Thank You.
Does Compact Hausdorff imply completely normal.
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general-topology
1 Answers
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The standard counterexample is the Tychonoff plank $[0,\omega_1]\times [0,\omega]$. The space is compact since is a product of compact spaces but the subspace obtained by removing the point $(\omega_1,\omega)$ is not normal.