der Dom
cathedral
A very large and important church that is often the seat of a bishop; a cathedral.
Der Kölner Dom ist ein berühmtes Wahrzeichen Deutschlands.
Cologne Cathedral is a famous landmark of Germany.

cathedral / fair
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cathedral
A very large and important church that is often the seat of a bishop; a cathedral.
Der Kölner Dom ist ein berühmtes Wahrzeichen Deutschlands.
Cologne Cathedral is a famous landmark of Germany.

The preposition often reveals the meaning: 'in den Dom gehen' (to go into the cathedral) refers to the church. 'Auf den Dom gehen' (to go to the fair), on the other hand, refers to the fair.
Die Touristen wollen den berühmten Dom von innen besichtigen.
The tourists want to see the famous cathedral from the inside.
In the meaning of 'Kathedrale' (cathedral), 'Dom' is often a fixed part of proper names of famous churches, such as the 'Kölner Dom' (Cologne Cathedral), the 'Ulmer Dom' (Ulm Minster), or the 'Stephansdom' (St. Stephen's Cathedral) in Vienna.
Wir haben bei unserem Besuch in Mailand den Dom besichtigt.
During our visit to Milan, we toured the cathedral.
The meaning 'Jahrmarkt' (fair) is regional and especially common in northern Germany. The best-known example is the 'Hamburger Dom' (Hamburg fair), one of the largest folk festivals in Germany.
Auf dem Bremer Dom gibt es viele Fahrgeschäfte und Essensstände.
At the Bremen fair, there are many rides and food stalls.
When 'Dom' (dome) is used in the sense of 'Kuppel' (dome), you almost always find the word in an architectural or technical context. It describes the hemispherical roof structure of a building.
Der Dom des Reichstagsgebäudes in Berlin ist für Besucher zugänglich.
The dome of the Reichstag building in Berlin is accessible to visitors.
church
'die Kirche' and 'der Dom' can both mean a Christian worship building. 'Kirche' is the more general word and works for both small and large buildings. 'Dom' is used for a particularly large and important church, often with a special history or as the church of a bishop. Learners may confuse the words because every 'Dom' is a church, but not every 'Kirche' is a 'Dom'. If you are unsure, 'Kirche' is usually the safe choice.

Für die Stadtführung treffen wir uns vor dem Dom.
For the city tour, we are meeting in front of the cathedral.
Am Sonntag geht die Familie in die Kirche am Marktplatz.
On Sunday, the family goes to the church on the market square.