He wants to close a gap between his teeth

The basic principle with regard to cosmetic procedures that
are done to remove a fault or deformity are that they are permissible,
because Islam is a religion of ease that aims to make things easy for
people. 

It was narrated that ‘Arfajah ibn As’ad (may Allaah be
pleased with him) said: My nose was cut off on the day of al-Kulaab during
the Jaahiliyyah (a famous battle) and I had a nose made of silver, but it
began to smell bad, so the Messenger of Allaah (peace and blessings of
Allaah be upon him) told me to get a nose made of gold. Narrated by
al-Tirmidhi (1770)’ classed as hasan by al-Nawawi in al-Majmoo’
(1/245)and by al-Albaani in Saheeh al-Tirmidhi. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

Beautification is of two types: beautification that is aimed
at removing a fault due to an accident etc – there is nothing wrong with
this, because the Prophet (peace and blessings of Allaah be upon him)
told a man whose nose had been cut off in battle to get a nose made of
gold. 

The second type is aimed at increasing beauty, which is not
aimed at removing a fault, rather the aim is to make someone more beautiful.
This is haraam and is not permissible because the Messenger (peace and
blessings of Allaah be upon him) cursed the woman who plucks eyebrows, the
one who has that done, the woman who does tattoos and the woman who has that
done, the woman who adds hair extensions and the woman who has that done,
because this is aimed at increasing beauty and is not removing a fault. 

Fatawa ‘Ulama’ al-Balad al-Haraam
(p. 575). 

A large gap between the teeth is an abnormality which some
people are affected by. It is permissible for one who has such a fault in
his teeth, or other faults, to correct it, and there is no sin on him for
doing that, in sha Allaah. 

Al-Nawawi (may Allaah have mercy on him) said in Sharh
Muslim (13/107): 

With regard to the phrase “women who file teeth”, what is
meant is doing that for the purpose of beautification. This indicates that
that which is haraam is that which is done for the purpose of
beautification. But if there is a need for that for medical reasons or to
correct a fault in the teeth etc, there is nothing wrong with that. And
Allaah knows best. End quote. 

The Standing Committee for Issuing Fatwas (24/75) was asked:
If I have two teeth that are long, is it permissible for me to make them
level with the other teeth? 

They replied: 

If they are so long that they bother you, they you may remove
that which is bothering you only. End quote. 

We have explained that it is permissible to do operations on
the teeth that will correct faults and abnormalities. Please see question
no. 1006 and
21255. 

And Allaah knows best.