How to withdraw from an application process in a polite manner? Say a candidate has met with a potential PhD supervisor and it was rather evident that he would be willing to take on the candidate. You have emailed a lot and met in person a few times, but nothing has been formalised yet. 

Suppose now the candidate decides to take it into a different direction simply because there are better options. 

What is the most polite way to withdraw ? Is it polite to send an email and explain, or does this come across as arrogant since a PhD candidate is not really in the position to turn down a Professor ? Would it be better simply not to get in touch again ? Is there some standard etiquette in the US and/or the UK ?
Just send a polite email stating that you found another position or that you simply prefer to take a different direction. Be sure to thank the professor for the time he has put into you. The email does not have to be long, but it should give a reason. You do not need to be overly elaborate, though. And don't forget to be polite.

The thing you do not realise is that the professor wants to find a student who will be committed to his topic for several years to come. If you are not such a student, it is best for the professor to know that before taking you on as a student. In a sense, you are doing the professor a favour by withdrawing.