If the crescent for Ramadan is viewed by non-Muslims but relayed live by TV, then should we Muslims after seeing it on TV accept it as having sighted the crescent and either begin the month or end it?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

If you mean seeing the Crescent by non-Muslims, we state that seeing the Crescent of the new lunar month should be done by a Muslim since sighting by a non-Muslim is not considered. The Prophet (Sallallahu Alaihi wa Sallam) said to a Bedouin who saw the Crescent: "do you bear the witness that La Ilaha Illa Allah (none deserves to be worshipped but Allah) and I am His Messenger?" The Bedouin replied in positive. So the Prophet (Sallallahu Alaihi wa Sallam) ordered people to start fasting. [ Abu Dawood, al-Nasaee and al-Tirmizi ]
Therefore, the Muslim jurists make a condition that the one who gives witness of seeing the Crescent should be a Muslim.
Then, it is recorded in al-Mudawana (a book of Maliki Fiqh): 'If two Muslim men give witness concerning seeing the Crescent, you have to start fasting (or said, to break fasting).'
If non-Muslim mass media convey the news concerning seeing the Crescent, such news is also not considered in Shari'a, since Muslim scholars refuse the witness of a dissolute person in religious matters. This means it is normal to refuse the witness of a non-Muslim. al-Zila'I in his book Nasb al-Rayah said, witness of a dissolute person is not accepted in the religions'.
According to al-Sheikh Muhammad Saleh Ibn al-Uthaimeen (may Allah's mercy be upon him), Muslims who live in a non-Muslim country are to do their best to see the Crescent if possible; otherwise, they are to do in accordance with what a Muslim country does concerning seeing the Crescent or not. Allah knows best.