
die Allgemeinheit
public
The entirety of people in a society; the general public.
Der neue Stadtpark ist für die Allgemeinheit frei zugänglich.
The new city park is freely accessible to the public.
public / generality
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public
The entirety of people in a society; the general public.
Der neue Stadtpark ist für die Allgemeinheit frei zugänglich.
The new city park is freely accessible to the public.

generality
The quality of being general and not specific; generality.
Die Allgemeinheit seiner Aussagen machte es schwer, ihm zu widersprechen.
The generality of his statements made it hard to contradict him.
The word is made up of the adjective 'allgemein' (general) and the ending '-heit'. The ending '-heit' forms feminine nouns that describe a quality or state.
Die Sicherheit der Allgemeinheit hat oberste Priorität.
The safety of the public has top priority.
The meaning can be recognized from context: If the word refers to people or society, it means 'die Öffentlichkeit' (the public). If it refers to the quality of a statement or rule, it means 'die Generalität' (generality).
Diese Regelung dient dem Schutz der Allgemeinheit.
This regulation serves to protect the public.
When 'Allgemeinheit' means the public, it is often used with the preposition 'für' (for). Typical phrases are 'für die Allgemeinheit zugänglich sein' (to be accessible to the public) or 'im Interesse der Allgemeinheit handeln' (to act in the public interest).
Der Strand ist für die Allgemeinheit geöffnet.
The beach is open to the public.
In the meaning of 'Generality', 'Allgemeinheit' is often used with a genitive that specifies what is general. Examples are 'die Allgemeinheit einer Regel' (the generality of a rule) or 'die Allgemeinheit einer Behauptung' (the generality of a claim).
Die Allgemeinheit des Gesetzes sorgt für viele Diskussionen.
The generality of the law is causing many discussions.
public
Die Öffentlichkeit and die Allgemeinheit can both refer to the people in a society. Öffentlichkeit emphasizes them more as recipients of information disseminated publicly, while Allgemeinheit emphasizes them as the whole group to which a measure applies or from which it benefits. The boundaries are not always clear-cut.

Die neue Schutzmaßnahme soll der Allgemeinheit dienen.
The new protective measure is intended to benefit the public.
Die Stadt informierte die Öffentlichkeit über die neue Schutzmaßnahme.
The city informed the public about the new protective measure.