Okay because of the inherent controversialness of the question, I will ask that before answering, that you state your opinion on people who allege to be transsexuals.
I am a believer that you should love everyone, and if a person who was born with a penis feels that they are a woman... then they are a woman. It's not my place to question who they are or how they live. If they want to use the women's washrooms, as long as they are not doing anything creepy... they can go ahead and go in.
However, the actual sex of a person does have an impact on their physical abilities. I mean even if we ignore the difference between endurance abilities and upper body vs lower body strength, which can be argued as nurture instead of nature (though I reject), quite simply each sex has parts that could create an advantage if we didn't have (I am looking specifically at swimming where even the difference of shaving your arms can make the difference in your speed... I would assume not having breasts would give an advantage to male-sexed people in this field).
So do you believe that a persons right to express themselves as the gender they are psychologically, extends to sports where physical attributes can actually create an advantage?