Cech added: "They've been there in the last few years so now I think they have to make sure they win something to show the progression".
"Success is ultimately winning trophies".
"Last year was a unusual season, we had more points but we finished outside of the top four".
Arsenal can pull within a point of Tottenham on Saturday when they meet at the Emirates, but defeat would see their season plunged into something of a crisis already.
"Three years ago, when we finished above them, we were disappointed because we didn't win the title".
"There is always one odd year where things can change".
And Cech thinks Tottenham made huge strides when appointing the Argentine.
"Tottenham have also been impressive and I had thought they would struggle going for the title with the Wembley and playing in the Champions League but they've gotten over it really well, so I'd have them third in the mix".
Most of Mauricio Pochettino's star players will have had breaks from football during the global fixtures, with the likes of Harry Kane, Dele Alli, Jan Vertonghen and Hugo Lloris all picking up injuries that kept them out of contention for their countries. His work at Southampton showed he is a very capable manager.
"They are an important part of the team", he added.
"They train well and we hope their form will help us reach the target".