
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.

to admit / approve
To officially accept or register someone or something for a specific purpose after an examination or review, for example for university studies, an exam, or road traffic.
Dieses Auto ist nicht für den Straßenverkehr zugelassen.
This car is not approved for road traffic.

to let in
To emotionally or mentally accept something or let it reach you, often feelings or thoughts.
Nach dem Streit fiel es ihm schwer, wieder Nähe zuzulassen.
After the argument, he found it difficult to let closeness in again.

to keep closed
To keep something closed or not open it.
Lass bitte das Fenster zu, es zieht.
Please keep the window closed; there's a draft.
'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 record / take in
A university zulässt someone for study when, after reviewing their application, it officially grants them permission to study. It aufnimmt the person when it actually integrates them into the university or a group of students.

Die Hochschule lässt Lea nach Prüfung ihrer Unterlagen zum Studium zu.
The university admits Lea to the program after reviewing her documents.
Die Hochschule nimmt Lea zum Wintersemester als Studentin auf.
The university admits Lea as a student for the winter semester.