zulassen
to allow
To allow, permit, or approve something. It means that you do not put an obstacle in the way or that you grant official permission.
Die Hausordnung lässt keine Haustiere zu.
The house rules do not allow pets.

to allow / admit / keep closed
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to allow
To allow, permit, or approve something. It means that you do not put an obstacle in the way or that you grant official permission.
Die Hausordnung lässt keine Haustiere zu.
The house rules do not allow pets.

'Zulassen' (to allow / admit / keep closed) is a separable verb. The prefix 'zu' stands at the end in a main clause, but remains connected to the verb in a subordinate clause. In the infinitive with 'zu', 'zu' stands between the prefix and the verb.
Er lässt keine Kritik zu.
He doesn't allow any criticism.
The prefix 'zu' has two opposite functions here. In most meanings, it means 'erlauben' (to allow) or 'ermöglichen' (to make possible). In the colloquial meaning 'geschlossen halten' (to keep closed), 'zu' means the opposite, namely 'closed'.
Wir können nicht zulassen, dass das passiert.
We cannot allow that to happen.
In this meaning, 'zulassen' (to admit / approve) is almost always used in an official or technical context. It is about formal approval after an examination or review.
Er wurde zum Medizinstudium zugelassen.
He was admitted to medical school.
When it is about feelings, 'zulassen' (to let in) describes the decision to lower an emotional barrier. Typical objects are abstract nouns such as 'Nähe' (closeness), 'Gefühle' (feelings), or 'Trauer' (grief).
Sie muss lernen, ihre Gefühle zuzulassen.
She has to learn to allow her feelings.
The meaning 'geschlossen halten' (to keep closed) is used almost only in spoken language and in direct requests or commands. The verb 'lassen' (to let / leave) often appears in the imperative here.
Kannst du bitte das Fenster zulassen?
Could you please keep the window closed?
to let / to have
'lassen' and 'zulassen' can both mean that something is allowed or not prevented. 'lassen' is very natural in everyday language, especially when one person allows another person to do something. 'zulassen' often sounds somewhat more formal or general. It fits well with rules, decisions, or situations in which you make something possible or do not prevent it. Learners may confuse the words because both have to do with permission. Remember: 'Ich lasse dich gehen' (I let you go) sounds personal and direct; 'die Regel lässt etwas zu' (the rule allows something) or 'jemand lässt etwas zu' (someone allows something) sounds more like deliberate permission or tolerance.

Herr Schmidt lässt im Park keine laute Musik zu.
Mr. Schmidt does not allow loud music in the park.
Sie lässt ihren Sohn nach den Hausaufgaben noch eine Stunde spielen.
She lets her son play for another hour after his homework.