# Weak Maximum Principle as a corollary of Strong Maximum Principle?

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 $$\overline\Omega$$?