While there was only one true Batman in each Gotham City (there were others who attempted to claim the mantle which we'll get to another time), there was several batwomen who aided the Dynamic Duo.
But again, only one Batwoman, aka Kathy Kane. Retired from the circus as a stunt-cyclist and trapeze artist who gained wealth, she patterned her new life after Batman. Become his female facsimile, she added her own touch to crimefighting, even acquiring a sidekick in her nice Betty Kane aka Batgirl.
On Earth-One she retired and ran her own circus, until she came out of retirement briefly to once more become Batwoman. Subsequently, she lost her life during a conflict with the Bronze Tiger. On Earth-Two, she ended up marrying and having children, coming out of retirement once more to aid her world's Robin and Earth-One's Batman in taking down Hugo Strange.