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I am reading http://www.axler.net/HFT.pdf. The author proves the following :

$\Omega$ is an open connected set and $u$ is real-valued and harmonic in $\Omega$. If $u$ attains a maximum in $\Omega$ then $u$ is constant.

After that he states the following as a corollary to this :

$\Omega$ is bounded and $u$ is real-valued on $\overline\Omega$ and harmonic on $\Omega$. Then $u$ attains its maximum over $\overline\Omega$ on $\partial\Omega$.

where connectedness of $\Omega$ has intentionally been omitted.

If we do not have connectedness, how can I possibly use the former result? Do I apply it on the connected components of $\overline\Omega$?

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