Maybe, but being in the drop zone by 1 point at the beginning of February isn't a disaster, and potentially losing against Liverpool would be more or less expected. It's the following 4 or 5 games that could define the season.
Saints article in the Guardian too ....
https://www.theguardian.com/football/blog/2018/jan/12/why-stale-southampton-have-lost-their-way-under-mauricio-pellegrino
A lot of negativity here today, and given recent performances it's hardly surprising but that was a decent half from Saints. I'm sure most here would have taken 0-0 at half time.