Helleofabore - great posts.
I am a bloke who is also a qualified personal trainer. I have been mad about and played sports since I was very young.
The worst thing people are tending to do nowadays, is to conflate gender and sex. Sports should be categorised primarily by sex, meaning male and female, it is that simple, or should be.
The case of Caster Semenya has been mentioned and some people still seem to be unaware that Caster and the other two athletes on the women's 800m podium at Rio 2016, all had the same 46XY DSD, meaning they are male. What many people are unaware of is that the IOC/IAAF were aware of this since 2009.
The other important event was the IOC Concensus Meeting 2015, which laid down the rules for intersex and transgender athletes competing in the women's category. They were IMHO unduly influenced by a deeply flawed study carried out by one Joanna Harper.
They focussed entirely on Testosterone levels, totally ignoring the massive differences in physiology between males and females.
Males have larger hearts and lungs, plus more oxygen producing red blood cells, leading to a more efficient cardio vascular syste. Males have longer, more dense bone structure, they have less fat and leaner muscle mass. They are able to build muscle more effectively than females.
Therefore allowances have to be made for female sport. In throwing events, the shot, hammer, discus and javelin are all lighter. The hurdles are lower. In cricket, women cricketers use lighter bats, smaller, lighter balls. In volleyball the net is lower.
In most athletic events, there is roughly a 10% difference between men's and women's records.
Roughly fifteen thousand men have run faster than the women's marathon record and sixteen year old boys have run faster than the women's 100m record.
stillmed.olympic.org/Documents/Commissions_PDFfiles/Medical_commission/2015-11_ioc_consensus_meeting_on_sex_reassignment_and_hyperandrogenism-en.pdf
Moving away from sport, there are obvious reasons (to anyone with an ounce of empathy) why women need their own safe spaces, where males will not be present - toilets, changing rooms, hospital wards, refuges, prisons, recreational facilities where they can feel safe.
Why do I say "obvious reasons"?
Because women face violence from men. The vast majority of violent offenders in prisons are men as are the vast majority of domestic abusers.
I don't really care how someone identifies as long as they are not causing distress to anyone else.
Males who are trying gain access to female safe spaces, by identifying as women are going to cause fear and distress to the women in those spaces.