City Councilwoman Nancy McFarlane, real estate executive Billie Redmond and obstetrician Randall Williams were the only candidates to file for the mayoral race when the candidate filing period ended at noon Friday.
Meeker announced in May that he would not seek a sixth two-year term as mayor.
If no candidate receives more than 50 percent of the votes in the Oct. 11 election, a runoff between the top two finishers will be held Nov. 8.
In Cary, Mayor Harold Weinbrecht faces competition in his re-election bid. Michelle Muir is also seeking the seat.