Tottenham finished ahead of Arsenal for the first time in 22 years when they claimed second spot in the Premier League last season, while the Gunners missed out on a place in the Champions League by ending the campaign in fifth place.
Asked if there had been a power shift in north London, Cech said: "No I don't think so. There is always one odd year where things can change, but the most important thing for us is competing against the other teams for the Premier League title and for the glory.
"We are not competing against Tottenham Hotspur. Two years ago when we finished above them we were disappointed because we didn't win the title.
"Our target is to win the title and be successful during the season, not necessarily looking at what Tottenham are doing."
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. We were under pressure the last couple of years, people talking about winning trophies, obviously we are disappointed not to have won the title but we won FA Cups. Although the season was not always what we wanted, we always had a trophy at the end of it."