Gender is a social construct. Sex is a medical definition. Gender can be fluid and ever changing. Sex is fixed at birth.
This is obvious, but it is at the crux of every debate that is happening surrounding transgender people and so I really think people are struggling with the definitions. Or, more likely, it is all ideological lines in the sand.
Can trans women compete in women’s sports? Well, why do women’s sports exist in the first place? They exist because there are phisological differences between males and females that render competition unfair in most all athletic contests. Each college and pro sports would be dominated by one sex. Most sports would be exclusively male, though gymnastics would likely be female dominated. If gender is fluid anyone could switch between the two at any time.
Will people do this? Probably not.
Will suddenly trans women take over womens’s sports if allowed? No. Almost certainly not, but it does kind of beg the question of why the category exists at all.
People should be able to be themselves and do what they want as long as they aren’t infringing on others rights. And trans people are most certainly included in that. However, some people feel that transwomen are infringing upon women’s rights laws such as title ix. The answer to that lies in the question of are sports divided by gender or sex. If by gender every gender, of which there are many, would have equal claim to the same rights under title ix. This would be an impossible requirement to fill. The intention of the law was always to be divided by sex, not gender, even if historically we, as a society, have used the terms interchangeably.