Back during the Republican presidential race, when Mayor Giuliani and Governor Romney were battling it out for the nomination, Mr. Giuliani claimed an indicator of his success was that he was followed in office by a Republican he helped get elected, while Mr. Romney was followed by a Democrat. As a political matter, the claim did not get Mr. Giuliani far, but as a substantive insight it was right on point.

Mayor Bloomberg not only consolidated but even improved on Mr. Giuliani’s gains in crime reduction, welfare reform, and economic growth in New York City, on the latter point with notable help from President Bush. Had those gains instead been reversed by a successor hostile to Mr. Giuliani, the legacy of Mr. Giuliani would have been the worse for it, and, more importantly, we would be living in a different city today.

Now Mr. Bloomberg is approaching the autumn of his mayoralty career. He is more than halfway done with the second of the two consecutive terms he is allowed under the city charter. And the question that is the source of rising concern in the city’s establishment is who will carry on Mr. Bloomberg’s legacy once the current mayor leaves office.

The natural candidate would be someone from Mr. Bloomberg’s own administration. The name at the top of some polls is the police commissioner, Raymond Kelly. But Mr. Kelly has not signaled his interest in the mayoralty. If he did he’d get everyone’s attention. But there are no term limits on the job of police commissioner, and Mr. Kelly is such a fine one that it is possible that he could best serve the public in that job in the next administration.

As for those candidates who have signaled their interest, the field is not one that could be expected to extend the Bloomberg legacy. Rep. Anthony Weiner ran against Mr. Bloomberg in 2005 and then endorsed Mr. Bloomberg’s opponent, Fernando Ferrer. Mr. Weiner is a former aide to Senator Schumer, who this week decided to launch an assault on Mr. Bloomberg, calling the mayor’s plan for the West side of Manhattan the “goofiest thing I’ve ever seen.” Mr. Weiner likes to proclaim at dinners of the Zionist Organization of America that he represents “the ZOA wing of the Democratic Party,” but his call for abandoning the cause of freedom in Iraq has been breathtaking. Meanwhile he’s been opposing the Bush tax cuts that have been such a boon to the New York economy.

The City Council speaker, Christine Quinn, is one of the most charming and likeable pols, but she has been badly damaged by the Council slush fund scandal. The comptroller, William Thompson Jr., is another likeable Democrat, but, while he mightn’t be culpable in the slush fund scandal, he failed to detect it. He has, in any event, also allowed the city’s pension funds to be represented by trial lawyers who have helped underwrite his campaigns. The chatter in lower Manhattan is that Mr. Thompson is walking rather than running for mayor.

There is a billionaire businessman, John Catsimatidis, but he has yet to articulate a rationale for his campaign and is running for the Republican nomination although his best-known political role heretofore has been as a fundraiser for and crony of the Clintons. It would be shortsighted to forget, or underestimate, the president of Brooklyn, Marty Markowitz, who has a genial flair but whose mettle has yet to be really tested and who, in any event, has not really emerged in the policy dispute at the serious levels at which any mayor has to be engaged.

One name that we are starting to hear talked about is that of the schools chancellor, Joel Klein. The chancellor made his national reputation as a trust-buster in President Clinton’s Justice Department, where he challenged Microsoft. At the education department here, Mr. Klein has begun to emerge as an advocate of parental choice via the charter school movement. Unlike these columns, he hasn’t become an advocate of full parental choice via a Milton Friedman-esque system of school vouchers. But if he were to enter the mayor’s race flying the flags of a trust-buster on schools and the teachers union, it would be a campaign that, if successful, could cement Mr. Bloomberg’s legacy.

Certainly someone should try – and the mayor would be wise to encourage whoever it is. Mr. Bloomberg has done a lot to improve this city. But the current slowing of economic growth means the building boom that — with the Javits Convention Center expansion, Moynihan Station, Atlantic Yards, Hudson Yards, ground zero, the Con Edison site — would have been a centerpiece of Mr. Bloomberg’s great legacy will be completed more slowly than expected, if at all. All the other gains under Mr. Bloomberg, in education, development, and crime reduction, are also reversible. So for all that Mr. Bloomberg has done for the city, the biggest may be to get behind one or several of the candidates who can be trusted to fight for his policies and vision. If he can, as Mr. Giuliani did, find a capable heir, his own legacy will be the better for it, as will the city.